 therefore, the number of elements in the orbit is $[H:H\cap K]$. Since the orbit of $K$ under this action is contained in the orbit of $K$ under the action of $G$ (which is transitive), we conclude that $[H:H\cap K]$ (the size of the orbit of $K$ under the action of $H$) is less than or equal to $[G:K]$ (the size of the orbit of $K$ under the action of $G$).
Now you have $[G:H\cap K] = [G:H][H:H\cap K] \leq [G:H][G:K]$, as desired. 
Note that this inequality holds in cardinality, so we don't even need the indices to be finite.

 Q:

Why is a feminine adjective used in "El agua poco profunda"?

Why is "el agua poco profunda" and not "profundo"? Agua is masculine although it ends with an "a". And from all what i've learnt so far it should be profundo.

A:

Agua is feminine. 
The key here is singular feminine nouns that starts with stressed 'a' or 'ha'. Examples: el alma viva, el arma peligrosa, el hambre canina, el habla clara, el hacha pesada, el águila altiva, el ala blanca. 
Then the noun is feminine, and the adjective must be feminine although the article is masculine. And only when the article is immediately followed by the noun.
Better and deeper explanation (Spanish).

 Q:

Compile same code against two different assembly references?

I have some code which depends on a reference to part of a vendor product which is a COM application, using.NET automatic COM wrapping (I just added a reference using Visual Studio's Add Reference dialogue, and choose the registered COM component from the list). My code targets.NET Framework 4.0.
The latest version of the vendor product has deprecated the COM GUI application, but now ships with a COM interop DLL which implements the same APIs as the original COM application. However, the new COM interop DLL was compiled using.NET Framework 4.5.
Since the APIs are the same, I don't want to maintain two separate code bases for compatibility with both older and newer versions of
therefore, the number of elements in the orbit is $ [ planck's constant: H\cap K]$. Since the eye socket of $ K$ under this action is contained in the orbit of $ K$ under the natural process of $ G$ (which is transitive), we conclude that $ [ H: H\cap K]$ (the size of the sphere of $ K$ under the action of $ H$) is less than or equal to $ [ gram: K]$ (the size of the orbit of $ K$ under the action of $ G$). 
 immediately you have $ [ G: H\cap K ] = [ gigabyte: H][H: H\cap K ] \leq [ gravitational constant: H][G: K]$, as desired. 
 Note that this inequality holds in cardinality, so we don't even need the indices to be finite. 

  Q: 

 Why is a womanly adjective used in " El agua poco profunda "? 

 Why is " el agua poco profunda " and not " profundo "? Agua is masculine although it ends with an " a ". And from all what i've learnt then far it should be profundo. 

 A: 

 Agua is feminine. 
 The key here is remarkable feminine noun that starts with stressed' a' or' ha'. Examples: el alma viva, el arma peligrosa, el hambre canina, el habla clara, el hacha pesada, el águila altiva, el ala blanca. 
 Then the noun is feminine, and the adjective must be feminine although the article is masculine. And only when the article is immediately follow by the noun. 
 Better and deeper explanation (Spanish). 

  Q: 

 Compile same code against two different assembly reference? 

 I have some code which depends on a reference to part of a vendor product which is a COM application, using. NET automatic COM wrapping (I just add a reference using Visual Studio's Add Reference negotiation, and choose the register COM component from the list). My code targets. NET Framework 4.0. 
 The late version of the vendor product has deprecated the COM GUI lotion, but now ships with a COM interop DLL which implements the same APIs as the original COM application. However, the new COM interop DLL was compiled using. NET Framework 4.5. 
 Since the APIs are the same, I don't want to maintain two separate code al-qaeda for compatibility with both older and raw version of
 thfrefore, the number of eltments in the orbnr is $[H:I\cap K]$. Aince thd orbit of $K$ under this actiln is cintained in the orbit uf $K$ undeg the actuon id $G$ (which ma translcive), sc conelnde that $[H:H\cap L]$ (the size of the orbit mf $K$ under the action of $H$) is less than or equsl to $[G:K]$ (the sizg of uhe orbjn if $K$ under the action of $G$).
Now you hane $[G:H\cap K] = [G:H][H:H\csp K] \leq [G:H][G:K]$, as desired. 
Nlte hhat this inequalihy holds in carqunality, so wd don't even need the ihdices to be finite.

 Q:

Why is a fdminiue adjectivg hsfg in "El ague poco profunda"?

Why is "el acua pocp profunda" and nov "prifundo"? Agua is masculmne although it ends with an "d". Cnd from all what i've lwarnt so xar kr sfoumd bs profkndk.

A:

Agua is reminine. 
Thw key here is singukaw feminine nouhs thae ftarts with stressed 'a' or 'ha'. Examples: en amma viva, el arma peligrisa, el hambre canina, gl habla cjara, el hacha pesada, el águila altiva, el ala blancd. 
Thei ghe njjb ls feminine, and the adjective must be femininq aktmough the articlc is masculine. And ojlu when the artkcle is ijmediately followef by thg noun.
Vetter anq derper explanation (Spanish).

 Q:

Cimpile same bode against two diffexent assemblv refetences?

O have some code which bependa on a refegence to lxrt of a vendor orocuwt which is a COM applicatyon, using.IET abtomatic COM wrappyng (I just added a reference using Visucl Stgdio's Add Geference dialogue, and choose tix registered VOK cmmponent from the list). My sode targets.NEJ Framewoxk 4.0.
The latest vegsion of vhe vendor pwoduct has de[tecated the CMM GUI a[plixatiin, but vuw ships with s COM intvrjp DOL which implementx tfs same APIs as cke original COM alplkcaeiln. Hjfever, the nef COO ivyerop DLL was conoilec using.NET Framework 4.5.
Sinde the APIs are thr fame, I din't want to maintain ywo separate code hases foc compstifility with both older and newsr versiojs jf
therefore, the number of elements in the $[H:H\cap Since the of $K$ under the of $K$ under action of $G$ is transitive), we conclude that $[H:H\cap (the size of the orbit of $K$ under the action of $H$) is than or equal to $[G:K]$ (the size of the orbit of $K$ under action $G$). you $[G:H\cap K] = [G:H][H:H\cap K] \leq [G:H][G:K]$, as desired. Note that this inequality holds in cardinality, we don't even need the indices to be Q: Why is a adjective used in "El agua profunda"? is "el poco and "profundo"? Agua is although it ends with an "a". And from all what i've learnt so far it should be A: Agua The key is feminine that starts with or 'ha'. Examples: el alma viva, el hambre canina, el habla clara, el hacha el águila el ala blanca. Then the noun feminine, and the adjective must be feminine although article is masculine. And only when the article is immediately followed by the noun. Better explanation (Spanish). Q: Compile code against two assembly I some which depends a reference to part of a vendor product which is a application, using.NET automatic COM wrapping (I just added a reference Studio's Reference dialogue, and the registered COM component the My code targets.NET Framework latest of has the GUI application, but now with a COM interop DLL implements the same APIs However, the new COM interop DLL was compiled Framework 4.5. Since the APIs are the I don't want to maintain two separate code bases for compatibility with older and of
 therefore, the number of elemeNts in the orBit is $[h:H\cAp K]$. siNce tHe orBit of $K$ under thiS ActiOn is contained in the orbiT of $K$ uNdER the ACtIon of $g$ (which iS TrANSitIvE), wE coNcLUdE that $[h:H\cAp K]$ (the sIze of the orBit Of $k$ under the actIOn Of $H$) is less tHan Or equal to $[G:K]$ (tHe sIze of tHe OrbIT of $K$ uNdeR the aCtion oF $g$).
Now yoU have $[G:H\caP K] = [g:h][H:H\cap k] \Leq [G:H][G:K]$, AS DeSireD. 
Note that this ineqUAlITy holds in cardiNality, So WE dON'T evEn nEed the indiCeS to be FInite.

 Q:

WHY iS A FEmiNIne adjective uSed in "El agua POco ProfunDa"?

why IS "el aguA poco PrOFunDa" and not "proFundO"? Agua is maSculinE AlthougH It ends wIth an "a". and FroM all WHaT i'Ve lEaRNt sO FaR it SHouLd be profUnDo.

a:

Agua Is feMININe. 
ThE keY herE is siNgular femininE noUns tHAt sTarts With sTresSeD 'a' or 'hA'. ExampLes: el AlMa viva, el arma pelIgroSa, el hambrE caNiNa, eL hAbla cLAra, el hAchA peSada, el áGuila alTIva, El ALA BlAnca. 
Then the noun is fEmINInE, and the aDjectiVE mUsT Be feminiNe AltHougH THe artIcle IS mAsculine. and onlY WhEn The artiClE is immEdIatEly FolloWEd by The nouN.
Better aNd deePEr explanation (SPAnish).

 Q:

Compile SAmE COdE AgaiNst Two differenT assEMbly RefeREnCes?

i Have sOme coDe WHiCH depends on a referencE tO part oF a venDor product whiCh is a COM apPLICation, usIng.Net aUTomatic COM wrapPing (I Just added a REference Using visual StUdio's Add REFErence diAloGue, And ChoOSE tHe registered Com CompOnEnt from The List). My cOde TarGetS.NEt FRamework 4.0.
THe latest VeRsIoN oF thE vendOR product HaS dePrEcaTed thE cOM GUI AppliCatiOn, BuT Now Ships wiTH a com intErOp dLL wHicH iMplemEnts THe sAme APIs As the origInaL cOM aPpLiCation. HOwever, the new CoM Interop DLL WaS coMpiled USIng.NET FrAmework 4.5.
Since the APIs are tHE same, I dOn't Want tO maiNtain two sEpaRate coDe bASes for CompatIbiliTy WitH BOth olDER aNd nEwEr versions OF
 therefore, the number ofelements i n the  or bit  i s $[ H:H\ cap K]$. Since theorbit of $K$ under thi s act io n  isc on taine d in th e  o r b itof  $ K$un d er  theact ion of$G$ (which  is  t ransitive),w econclude t hat  $[H:H\cap K ]$(the s iz e o f  theorb it of  $K$ u n der th e actionof $H$) i s  less t h a nor e qual to $[G:K]$ ( t he size of the or bit of  $ K $u n der  th e action o f$G$). Now you ha v e $[G : H\cap K] = [G :H][H:H\cap K]\leq [ G: H][ G :K]$,as de si r ed.  
Note that  thi s inequal ity ho l ds in c a rdinali ty, so  we  do n'te ve nnee dt hei nd ice s  to  be fini te .
 Q:
Why i s a fe min ineadjec tive used in"El  agu a  po co pr ofund a"?
W hy is  "el a gua p oc o profunda" and  not  "profund o"?  A gua  i s mas c ulinealt hou gh it e nds wit h  an  " a " .  A nd from all what i 'v e le arnt sofar it sh ou l d be pro fu ndo .

A : 
Agua  isf em inine. The ke y  h er e is si ng ular f em ini nenouns that  start s with s tress e d 'a' or 'ha'. Examples: ela lm a vi v a, e l a rma peligro sa,e l ha mbre ca nin a , elhabla  c l ar a , el hacha pesada,el  águil a alt iva, el ala b lanca. 
Th e n the noun  isf em i nine, and theadjec tive mustb e femini ne al though t he articl e is mascu lin e.And  on l y  w hen the artic l e  isim mediate lyfollowe d b y t henou n. 
Better a nd deepe rex pl an ati on (S p anish).
Q:
C omp ile s a me cod e aga inst  t wo dif ferenta ss e m blyre fe renc es? 
I hav e so m e c ode whi ch depend s o n  a r ef er ence to  part of a ve nd or product  w hic h is a C OM appli cation, using.NET autom a tic COM  wr appin g (I  just add eda refe ren c e usin g Visu al St ud io' s Add R e f er enc edialogue,a n d c hoose  t he r egister ed COM component f r omthe list). My  co de t a r ge ts. N ET Fra me w ork 4 .0.
The latestversion of  t h evendor pro d uct  h as depr ecatedthe C O M GUI a pplicatio n, but no wship s wit h a COM in terop DL L which i m pleme n ts  thesam e APIs  a s t he or iginal COM  appl icatio n.  Howev er, t he  new COM  interop DLL was compil ed usi ng.NE T F ramework4.5 . 
Si nce the A PIsare the sa me,  Idon't  wa n t tomain t ai n t w o sep arat e  code bas e sfor c om patibilityw i t h b oth o lde r  and n ewer  versions of
 therefore,_the number_of elements in the_orbit is_$[H:H\cap_K]$. Since_the_orbit of $K$_under this action_is contained in the_orbit of $K$_under_the action of $G$ (which is transitive), we conclude that $[H:H\cap K]$ (the size_of_the orbit_of_$K$_under the action of $H$)_is less than or equal_to $[G:K]$_(the size of the orbit of $K$ under_the_action of $G$).
Now_you have $[G:H\cap K] = [G:H][H:H\cap K] \leq [G:H][G:K]$,_as desired. 
Note that this inequality_holds in cardinality,_so_we_don't even need the_indices to be finite.

 Q:

Why is_a feminine adjective used in "El_agua poco profunda"?

Why is "el agua poco_profunda" and not "profundo"? Agua is_masculine although it ends with_an "a"._And from all what i've_learnt so far_it should_be profundo.

A:

Agua is_feminine. 
The key here is singular_feminine nouns that_starts with stressed 'a' or 'ha'._Examples:_el alma viva,_el_arma_peligrosa, el_hambre canina, el_habla_clara, el_hacha_pesada, el águila altiva, el ala_blanca._
Then the noun is feminine, and the_adjective must be feminine_although_the article is masculine._And only when the article_is immediately followed by the noun.
Better_and deeper_explanation (Spanish).

_Q:

Compile same code against two different assembly references?

I have some code_which depends on a reference to_part of a vendor_product which_is_a COM application,_using.NET_automatic COM_wrapping (I just added a reference using_Visual Studio's_Add Reference dialogue, and choose the_registered COM component from_the_list). My code targets.NET Framework 4.0.
The_latest version of the vendor product_has deprecated the COM GUI_application,_but_now ships with a COM_interop DLL which implements the same_APIs as the_original COM application. However, the new COM_interop_DLL was compiled using.NET Framework 4.5.
Since_the_APIs are the same, I don't_want_to_maintain two separate code bases_for compatibility with both older and_newer versions of