
AxAyAz(F(x,g(y,z)) ↔ h(x,y) = h(x,z)) :PR
Show AwAxAyAz(F(w,g(x,y))∧F(w,g(x,z)) → F(w,g(y,z)))
 Show AxAyAz(F(a,g(x,y))∧F(a,g(x,z)) → F(a,g(y,z)))
  Show AyAz(F(a,g(b,y))∧F(a,g(b,z)) → F(a,g(y,z)))
   Show Az(F(a,g(b,c))∧F(a,g(b,z)) → F(a,g(c,z)))
    Show F(a,g(b,c))∧F(a,g(b,d)) → F(a,g(c,d))
     F(a,g(b,c))∧F(a,g(b,d))             :AS
     F(a,g(b,c))                         :S 7
     F(a,g(b,d))                         :S 7
     AyAz(F(a,g(y,z)) ↔ h(a,y) = h(a,z)) :UI 1
     Az(F(a,g(b,z)) ↔ h(a,b) = h(a,z))   :UI 10
     F(a,g(b,c)) ↔ h(a,b) = h(a,c)       :UI 11
     F(a,g(b,d)) ↔ h(a,b) = h(a,d)       :UI 11
     F(a,g(b,c)) → h(a,b) = h(a,c)       :BC 12
     F(a,g(b,d)) → h(a,b) = h(a,d)       :BC 13
     h(a,b) = h(a,c)                     :MP 8 14
     h(a,b) = h(a,d)                     :MP 9 15
     h(a,c) = h(a,d)                     :LL 16 17
     Az(F(a,g(c,z)) ↔ h(a,c) = h(a,z))   :UI 10
     F(a,g(c,d)) ↔ h(a,c) = h(a,d)       :UI 19
     h(a,c) = h(a,d) → F(a,g(c,d))       :BC 20
     F(a,g(c,d))                         :MP 18 21
    :CD 22
   :UD 6
  :UD 5
 :UD 4
:UD 3|]

transitiveTheorem = [st|
Show P(a) within P(P(a))
 a within P(
bisectorTheorem = [st|
AxAyAz(F(x,g(y,z)) ↔ h(x,y) = h(x,z)) :PR
Show AwAxAyAz(F(w,g(x,y))∧F(w,g(x,z)) → F(w,g(y,z)))
 Show AxAyAz(F(a,g(x,y))∧F(a,g(x,z)) → F(a,g(y,z)))
  Show AyAz(F(a,g(b,y))∧F(a,g(b,z)) → F(a,g(y,z)))
   Show Az(F(a,g(b,c))∧F(a,g(b,z)) → F(a,g(c,z)))
    Show F(a,g(b,c))∧F(a,g(b,d)) → F(a,g(c,d))
     F(a,g(b,c))∧F(a,g(b,d))             :AS
     F(a,g(b,c))                         :S 7
     F(a,g(b,d))                         :S 7
     AyAz(F(a,g(y,z)) ↔ h(a,y) = h(a,z)) :UI 1
     Az(F(a,g(b,z)) ↔ h(a,b) = h(a,z))   :UI 10
     F(a,g(b,c)) ↔ h(a,b) = h(a,c)       :UI 11
     F(a,g(b,d)) ↔ h(a,b) = h(a,d)       :UI 11
     F(a,g(b,c)) → h(a,b) = h(a,c)       :BC 12
     F(a,g(b,d)) → h(a,b) = h(a,d)       :BC 13
     h(a,b) = h(a,c)                     :MP 8 14
     h(a,b) = h(a,d)                     :MP 9 15
     h(a,c) = h(a,d)                     :LL 16 17
     Az(F(a,g(c,z)) ↔ h(a,c) = h(a,z))   :UI 10
     F(a,g(c,d)) ↔ h(a,c) = h(a,d)       :UI 19
     h(a,c) = h(a,d) → F(a,g(c,d))       :BC 20
     F(a,g(c,d))                         :MP 18 21
    :CD 22
   :UD 6
  :UD 5
 :UD 4
:UD 3|]
bisectorTheorem = [st|
AxAyAz(F(x,g(y,z)) ↔ h(x,y) = h(x,z)) :PR
Show AwAxAyAz(F(w,g(x,y))∧F(w,g(x,z)) → F(w,g(y,z)))
 Show AxAyAz(F(a,g(x,y))∧F(a,g(x,z)) → F(a,g(y,z)))
  Show AyAz(F(a,g(b,y))∧F(a,g(b,z)) → F(a,g(y,z)))
   Show Az(F(a,g(b,c))∧F(a,g(b,z)) → F(a,g(c,z)))
    Show F(a,g(b,c))∧F(a,g(b,d)) → F(a,g(c,d))
     F(a,g(b,c))∧F(a,g(b,d))             :AS
     F(a,g(b,c))                         :S 7
     F(a,g(b,d))                         :S 7
     AyAz(F(a,g(y,z)) ↔ h(a,y) = h(a,z)) :UI 1
     Az(F(a,g(b,z)) ↔ h(a,b) = h(a,z))   :UI 10
     F(a,g(b,c)) ↔ h(a,b) = h(a,c)       :UI 11
     F(a,g(b,d)) ↔ h(a,b) = h(a,d)       :UI 11
     F(a,g(b,c)) → h(a,b) = h(a,c)       :BC 12
     F(a,g(b,d)) → h(a,b) = h(a,d)       :BC 13
     h(a,b) = h(a,c)                     :MP 8 14
     h(a,b) = h(a,d)                     :MP 9 15
     h(a,c) = h(a,d)                     :LL 16 17
     Az(F(a,g(c,z)) ↔ h(a,c) = h(a,z))   :UI 10
     F(a,g(c,d)) ↔ h(a,c) = h(a,d)       :UI 19
     h(a,c) = h(a,d) → F(a,g(c,d))       :BC 20
     F(a,g(c,d))                         :MP 18 21
    :CD 22
   :UD 6
  :UD 5
 :UD 4
:UD 3|]