:PR
 Ax(x in a -> x in P(a)):Def-S 2
 Show Ax(x in P(a) -> x in P(P(a)))
  Show b in P(a) -> b in P(P(a))
   b in P(a) :AS
   b within a:Def-P 6
   Ax(x in b -> x in a):Def-S 7
   Show b in P(P(a))
    Show Ax(x in b -> x in P(a))
     Show c in b -> c in P(a)
      c in b :AS
      c in b -> c in a:UI 8
      c in a :MP 12 13
      c in a -> c in P(a) :UI 3
      c in P(a):MP 14 15
     :CD 16
    :UD11
    b within P(a):Def-S 10
    b in P(P(a)):Def-P 19
   :DD 20
  :CD 9
 :UD 5
 P(a) within P(P(a)):Def-S 4
:DD 24|]

sequentDemo = [st|
 {
  "label": "AxEy(F(x)/\\G(y)):|-:EyAx(F(
transitiveTheorem = [st|
Show P(a) within P(P(a))
 a within P(a):PR
 Ax(x in a -> x in P(a)):Def-S 2
 Show Ax(x in P(a) -> x in P(P(a)))
  Show b in P(a) -> b in P(P(a))
   b in P(a) :AS
   b within a:Def-P 6
   Ax(x in b -> x in a):Def-S 7
   Show b in P(P(a))
    Show Ax(x in b -> x in P(a))
     Show c in b -> c in P(a)
      c in b :AS
      c in b -> c in a:UI 8
      c in a :MP 12 13
      c in a -> c in P(a) :UI 3
      c in P(a):MP 14 15
     :CD 16
    :UD11
    b within P(a):Def-S 10
    b in P(P(a)):Def-P 19
   :DD 20
  :CD 9
 :UD 5
 P(a) within P(P(a)):Def-S 4
:DD 24|]
transitiveTheorem = [st|
Show P(a) within P(P(a))
 a within P(a):PR
 Ax(x in a -> x in P(a)):Def-S 2
 Show Ax(x in P(a) -> x in P(P(a)))
  Show b in P(a) -> b in P(P(a))
   b in P(a) :AS
   b within a:Def-P 6
   Ax(x in b -> x in a):Def-S 7
   Show b in P(P(a))
    Show Ax(x in b -> x in P(a))
     Show c in b -> c in P(a)
      c in b :AS
      c in b -> c in a:UI 8
      c in a :MP 12 13
      c in a -> c in P(a) :UI 3
      c in P(a):MP 14 15
     :CD 16
    :UD11
    b within P(a):Def-S 10
    b in P(P(a)):Def-P 19
   :DD 20
  :CD 9
 :UD 5
 P(a) within P(P(a)):Def-S 4
:DD 24|]