%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU850^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n189.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:17 EDT 2014

% Result   : Timeout 300.08s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU850^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n189.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:36:36 CDT 2014
% % CPUTime  : 300.08 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x196eb48>, <kernel.Type object at 0x1b50b90>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (S:(a->Prop)) (T:(a->Prop)) (U:(a->Prop)), (((and (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U))->((and ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (forall (Xx:a), ((U Xx)->(S Xx)))))) of role conjecture named cGAZING_THM9_pme
% Conjecture to prove = (forall (S:(a->Prop)) (T:(a->Prop)) (U:(a->Prop)), (((and (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U))->((and ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (forall (Xx:a), ((U Xx)->(S Xx)))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (S:(a->Prop)) (T:(a->Prop)) (U:(a->Prop)), (((and (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U))->((and ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (forall (Xx:a), ((U Xx)->(S Xx))))))']
% Parameter a:Type.
% Trying to prove (forall (S:(a->Prop)) (T:(a->Prop)) (U:(a->Prop)), (((and (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U))->((and ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (forall (Xx:a), ((U Xx)->(S Xx))))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((and_rect0 ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl) as proof of (P b)
% Found (((conj0 b) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl) as proof of (P b)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl) as proof of (P b)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl) as proof of (P b)
% Found x00:=(x0 (fun (x1:(a->Prop))=> (x1 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x1:(a->Prop))=> (x1 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl) as proof of (P b)
% Found (((conj0 b) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl) as proof of (P b)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl) as proof of (P b)
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl)) as proof of (P b)
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl)) as proof of ((((eq (a->Prop)) T) U)->(P b))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P b)))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl))) as proof of (P b)
% Found ((and_rect0 (P b)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl))) as proof of (P b)
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) (P b)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl))) as proof of (P b)
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) (P b)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) iff_refl))) as proof of (P b)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((and_rect0 ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of a0
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: a0:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of a0
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: a0:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of a0
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: a0:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of a0
% Found x0:(((eq (a->Prop)) S) T)
% Instantiate: b:=(forall (P:((a->Prop)->Prop)), ((P S)->(P T))):Prop
% Found x0 as proof of a0
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym) as proof of (P a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym) as proof of (P a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym) as proof of (P a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym) as proof of (P a0)
% Found eq_trans1000:=(eq_trans100 U):((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(((eq (a->Prop)) S) U)))
% Found (eq_trans100 U) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->a0))
% Found ((eq_trans10 T) U) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->a0))
% Found (((eq_trans1 S) T) U) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->a0))
% Found ((((eq_trans (a->Prop)) S) T) U) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->a0))
% Found ((((eq_trans (a->Prop)) S) T) U) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->a0))
% Found ((((eq_trans (a->Prop)) S) T) U) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->a0))
% Found (and_rect00 ((((eq_trans (a->Prop)) S) T) U)) as proof of a0
% Found ((and_rect0 a0) ((((eq_trans (a->Prop)) S) T) U)) as proof of a0
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) a0) ((((eq_trans (a->Prop)) S) T) U)) as proof of a0
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) a0) ((((eq_trans (a->Prop)) S) T) U)) as proof of a0
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym) as proof of (P a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym) as proof of (P a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym) as proof of (P a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym) as proof of (P a0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found x00:=(x0 (fun (x1:(a->Prop))=> (x1 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x1:(a->Prop))=> (x1 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found ((eq_ref a) b) as proof of (((eq a) b) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym) as proof of (P a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym) as proof of (P a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym) as proof of (P a0)
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym)) as proof of (P a0)
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym)) as proof of ((((eq (a->Prop)) T) U)->(P a0))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P a0)))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym))) as proof of (P a0)
% Found ((and_rect0 (P a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym))) as proof of (P a0)
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) (P a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym))) as proof of (P a0)
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) (P a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) eq_sym))) as proof of (P a0)
% Found x00:=(x0 (fun (x1:(a->Prop))=> (x1 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x1:(a->Prop))=> (x1 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) x0) as proof of (P a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) x0) as proof of (P a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) x0) as proof of (P a0)
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) x0)) as proof of (P a0)
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) x0)) as proof of ((((eq (a->Prop)) T) U)->(P a0))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) x0)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P a0)))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) x0))) as proof of (P a0)
% Found ((and_rect0 (P a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) x0))) as proof of (P a0)
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) (P a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) x0))) as proof of (P a0)
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) (P a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) x0))) as proof of (P a0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((and_rect0 ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((and_rect0 ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x0:(((eq (a->Prop)) S) T)
% Instantiate: b0:=(forall (P:((a->Prop)->Prop)), ((P S)->(P T))):Prop
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found x0:(((eq (a->Prop)) S) T)
% Instantiate: b0:=(forall (P:((a->Prop)->Prop)), ((P S)->(P T))):Prop
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found x0:(((eq (a->Prop)) S) T)
% Instantiate: b0:=(forall (P:((a->Prop)->Prop)), ((P S)->(P T))):Prop
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((and_rect0 ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P0:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P0)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P0) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b0:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b0
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_trans0000:=(eq_trans000 U):((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(((eq (a->Prop)) S) U)))
% Found (eq_trans000 U) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found ((eq_trans00 T) U) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found (((eq_trans0 S) T) U) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found ((((eq_trans (a->Prop)) S) T) U) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found ((((eq_trans (a->Prop)) S) T) U) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found ((((eq_trans (a->Prop)) S) T) U) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found (and_rect00 ((((eq_trans (a->Prop)) S) T) U)) as proof of (P0 b0)
% Found ((and_rect0 (P0 b0)) ((((eq_trans (a->Prop)) S) T) U)) as proof of (P0 b0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 b0)) ((((eq_trans (a->Prop)) S) T) U)) as proof of (P0 b0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 b0)) ((((eq_trans (a->Prop)) S) T) U)) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b0:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b0
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x0:(((eq (a->Prop)) S) T)
% Instantiate: b0:=(forall (P:((a->Prop)->Prop)), ((P S)->(P T))):Prop
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found ((eq_ref a) b) as proof of (((eq a) b) b0)
% Found x0:(((eq (a->Prop)) S) T)
% Instantiate: b0:=(forall (P:((a->Prop)->Prop)), ((P S)->(P T))):Prop
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found x0:(((eq (a->Prop)) S) T)
% Instantiate: b0:=(forall (P:((a->Prop)->Prop)), ((P S)->(P T))):Prop
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found x00:=(x0 (fun (x1:(a->Prop))=> (x1 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x1:(a->Prop))=> (x1 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found x0:(((eq (a->Prop)) S) T)
% Instantiate: b0:=(forall (P:((a->Prop)->Prop)), ((P S)->(P T))):Prop
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind)) as proof of (P0 b0)
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind)) as proof of ((((eq (a->Prop)) T) U)->(P0 b0))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind))) as proof of (P0 b0)
% Found ((and_rect0 (P0 b0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 b0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 b0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind))) as proof of (P0 b0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found x0:(((eq (a->Prop)) S) T)
% Instantiate: b0:=(forall (P:((a->Prop)->Prop)), ((P S)->(P T))):Prop
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found x0:(((eq (a->Prop)) S) T)
% Instantiate: b0:=(forall (P:((a->Prop)->Prop)), ((P S)->(P T))):Prop
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found x0:(((eq (a->Prop)) S) T)
% Instantiate: b0:=(forall (P:((a->Prop)->Prop)), ((P S)->(P T))):Prop
% Found x0 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b0:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b0
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b0:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b0
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b0:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b0
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x00:=(x0 (fun (x1:(a->Prop))=> (x1 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x1:(a->Prop))=> (x1 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind) as proof of (P0 b0)
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind)) as proof of (P0 b0)
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind)) as proof of ((((eq (a->Prop)) T) U)->(P0 b0))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind))) as proof of (P0 b0)
% Found ((and_rect0 (P0 b0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 b0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 b0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) or_ind))) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of a0
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of a0
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((and_rect0 ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) eq_sym0) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) eq_sym0) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) eq_sym0) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) eq_sym0) as proof of (P0 b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) eq_sym0) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) eq_sym0) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) eq_sym0) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) eq_sym0) as proof of (P0 b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) eq_sym0) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) eq_sym0) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) eq_sym0) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) eq_sym0) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b0
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b0
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((and_rect0 ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))):((P1 (forall (Xx:a), ((U Xx)->(S Xx))))->(P1 (forall (Xx:a), ((U Xx)->(S Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))):((P1 (forall (Xx:a), ((U Xx)->(S Xx))))->(P1 (forall (Xx:a), ((U Xx)->(S Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))):((P1 (forall (Xx:a), ((U Xx)->(S Xx))))->(P1 (forall (Xx:a), ((U Xx)->(S Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x00:=(x0 (fun (x1:(a->Prop))=> (x1 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x1:(a->Prop))=> (x1 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found x00:=(x0 (fun (x1:(a->Prop))=> (x1 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x1:(a->Prop))=> (x1 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of (P0 b0)
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of ((((eq (a->Prop)) T) U)->(P0 b0))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 b0)
% Found ((and_rect0 (P0 b0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 b0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 b0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of (P0 a0)
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of ((((eq (a->Prop)) T) U)->(P0 a0))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 a0)))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found ((and_rect0 (P0 a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found x00:=(x0 (fun (x1:(a->Prop))=> (x1 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x1:(a->Prop))=> (x1 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of (P0 a0)
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of ((((eq (a->Prop)) T) U)->(P0 a0))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 a0)))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found ((and_rect0 (P0 a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found conj10:=(conj1 (((eq (a->Prop)) T) U)):((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->((and (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U))))
% Found (conj1 (((eq (a->Prop)) T) U)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found ((conj (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found ((conj (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found ((conj (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found (and_rect00 ((conj (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U))) as proof of (P0 b0)
% Found ((and_rect0 (P0 b0)) ((conj (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 b0)) ((conj (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 b0)) ((conj (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U))) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((and_rect0 ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((and_rect0 ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of a0
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of a0
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))):((P1 (forall (Xx:a), ((U Xx)->(S Xx))))->(P1 (forall (Xx:a), ((U Xx)->(S Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((and_rect0 ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))):((P1 (forall (Xx:a), ((U Xx)->(S Xx))))->(P1 (forall (Xx:a), ((U Xx)->(S Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))):((P1 (forall (Xx:a), ((U Xx)->(S Xx))))->(P1 (forall (Xx:a), ((U Xx)->(S Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b0
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a0
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b0)
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a0
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of a0
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((and_rect0 ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b0
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a0
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of b0
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a0
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found or_comm_i:=(fun (A:Prop) (B:Prop) (H:((or A) B))=> ((((((or_ind A) B) ((or B) A)) ((or_intror B) A)) ((or_introl B) A)) H)):(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A)))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (((or A) B)->((or B) A))):Prop
% Found or_comm_i as proof of a0
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))):((P1 (forall (Xx:a), ((U Xx)->(S Xx))))->(P1 (forall (Xx:a), ((U Xx)->(S Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))):((P1 (forall (Xx:a), ((U Xx)->(S Xx))))->(P1 (forall (Xx:a), ((U Xx)->(S Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref000:=(eq_ref00 P1):((P1 (forall (Xx:a), ((U Xx)->(S Xx))))->(P1 (forall (Xx:a), ((U Xx)->(S Xx)))))
% Found (eq_ref00 P1) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) P1) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) P1) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) P1) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))):((P1 (forall (Xx:a), ((U Xx)->(S Xx))))->(P1 (forall (Xx:a), ((U Xx)->(S Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))):((P1 (forall (Xx:a), ((U Xx)->(S Xx))))->(P1 (forall (Xx:a), ((U Xx)->(S Xx)))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (x1 (fun (x2:(a->Prop))=> (P1 (forall (Xx:a), ((U Xx)->(S Xx)))))) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found eq_ref000:=(eq_ref00 P1):((P1 (forall (Xx:a), ((U Xx)->(S Xx))))->(P1 (forall (Xx:a), ((U Xx)->(S Xx)))))
% Found (eq_ref00 P1) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) P1) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) P1) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) P1) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found eq_ref000:=(eq_ref00 P1):((P1 (forall (Xx:a), ((U Xx)->(S Xx))))->(P1 (forall (Xx:a), ((U Xx)->(S Xx)))))
% Found (eq_ref00 P1) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) P1) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) P1) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) P1) as proof of (P2 (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x00:=(x0 (fun (x1:(a->Prop))=> (x1 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x1:(a->Prop))=> (x1 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found x00:=(x0 (fun (x1:(a->Prop))=> (x1 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x1:(a->Prop))=> (x1 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 b0)
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of (P0 b0)
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of ((((eq (a->Prop)) T) U)->(P0 b0))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 b0)))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 b0)
% Found ((and_rect0 (P0 b0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 b0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 b0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 b0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of (P0 a0)
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of ((((eq (a->Prop)) T) U)->(P0 a0))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 a0)))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found ((and_rect0 (P0 a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found x00:=(x0 (fun (x1:(a->Prop))=> (x1 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x1:(a->Prop))=> (x1 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x1:(a->Prop))=> (x1 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))):(((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found (eq_ref0 (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((U Xx)->(S Xx)))) b1)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1) as proof of (P0 a0)
% Found (fun (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of (P0 a0)
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of ((((eq (a->Prop)) T) U)->(P0 a0))
% Found (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1)) as proof of ((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->(P0 a0)))
% Found (and_rect00 (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found ((and_rect0 (P0 a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found (((fun (P1:Type) (x0:((((eq (a->Prop)) S) T)->((((eq (a->Prop)) T) U)->P1)))=> (((((and_rect (((eq (a->Prop)) S) T)) (((eq (a->Prop)) T) U)) P1) x0) x)) (P0 a0)) (fun (x0:(((eq (a->Prop)) S) T)) (x1:(((eq (a->Prop)) T) U))=> ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))))) proj1))) as proof of (P0 a0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 b0)
% Found (((conj0 b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 b0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) b0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 b0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (T Xx))):((T Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (T Xx))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((conj00 ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found (((conj0 a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found ((((conj ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))) a0) ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (T Xx)))))) or_comm_i) as proof of (P0 a0)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x10:=(x1 (fun (x2:(a->Prop))=> (x2 Xx))):((T Xx)->(U Xx))
% Found (x1 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((T Xx)->(U Xx))
% Found (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((T Xx)->(U Xx)))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (x1 (fun (x2:(a->Prop))=> (x2 Xx))))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(U Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (forall (Xx:a), ((U Xx)->(S Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of (forall (Xx:a), ((T Xx)->(T Xx)))
% Found ((conj10 (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found (((conj1 (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found ((((conj (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx)))) (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx))))) (fun (Xx:a)=> (((eq_ref a) Xx) T))) as proof of ((and (forall (Xx:a), ((S Xx)->(T Xx)))) (forall (Xx:a), ((T Xx)->(T Xx))))
% Found x00:=(x0 (fun (x2:(a->Prop))=> (x2 Xx))):((S Xx)->(T Xx))
% Found (x0 (fun (x2:(a->Prop))=> (x2 Xx))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of ((S Xx)->(T Xx))
% Found (fun (Xx:a)=> (x0 (fun (x2:(a->Prop))=> (x2 Xx)))) as proof of (forall (Xx:a), ((S Xx)->(T Xx)))
% Found eq_ref000:=(eq_ref00 T):((T Xx)->(T Xx))
% Found (eq_ref00 T) as proof of ((T Xx)->(T Xx))
% Found ((eq_ref0 Xx) T) as proof of ((T Xx)->(T Xx))
% Found (((eq_ref a) Xx) T) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) as proof of ((T Xx)->(T Xx))
% Found (fun (Xx:a)=> (((eq_ref a) Xx) T)) a
% EOF
%------------------------------------------------------------------------------