%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU844^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n116.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 13:33:16 EDT 2014

% Result   : Timeout 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU844^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n116.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 11:36:11 CDT 2014
% % CPUTime  : 300.02 
% 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 0x22dc8c0>, <kernel.Type object at 0x22db320>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x22dc488>, <kernel.DependentProduct object at 0x22db680>) of role type named cS
% Using role type
% Declaring cS:(a->Prop)
% FOF formula (<kernel.Constant object at 0x22dc050>, <kernel.DependentProduct object at 0x22db638>) of role type named cT
% Using role type
% Declaring cT:(a->Prop)
% FOF formula ((forall (S0:(a->Prop)) (T0:(a->Prop)), (((eq (a->Prop)) S0) T0))->((and (forall (Xx:a), ((cS Xx)->(cT Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))) of role conjecture named cGAZING_THM8_pme
% Conjecture to prove = ((forall (S0:(a->Prop)) (T0:(a->Prop)), (((eq (a->Prop)) S0) T0))->((and (forall (Xx:a), ((cS Xx)->(cT Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((forall (S0:(a->Prop)) (T0:(a->Prop)), (((eq (a->Prop)) S0) T0))->((and (forall (Xx:a), ((cS Xx)->(cT Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx)))))']
% Parameter a:Type.
% Parameter cS:(a->Prop).
% Parameter cT:(a->Prop).
% Trying to prove ((forall (S0:(a->Prop)) (T0:(a->Prop)), (((eq (a->Prop)) S0) T0))->((and (forall (Xx:a), ((cS Xx)->(cT Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx)))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS 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 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 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), ((cT Xx)->(cS Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found ex_ind:(forall (A:Type) (F:(A->Prop)) (P:Prop), ((forall (x:A), ((F x)->P))->(((ex A) F)->P)))
% Instantiate: b:=(forall (A:Type) (F:(A->Prop)) (P:Prop), ((forall (x:A), ((F x)->P))->(((ex A) F)->P))):Prop
% Found ex_ind as proof of b
% 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 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 relational_choice:(forall (A:Type) (B:Type) (R:(A->(B->Prop))), ((forall (x:A), ((ex B) (fun (y:B)=> ((R x) y))))->((ex (A->(B->Prop))) (fun (R':(A->(B->Prop)))=> ((and ((((subrelation A) B) R') R)) (forall (x:A), ((ex B) ((unique B) (fun (y:B)=> ((R' x) y))))))))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), ((forall (x:A), ((ex B) (fun (y:B)=> ((R x) y))))->((ex (A->(B->Prop))) (fun (R':(A->(B->Prop)))=> ((and ((((subrelation A) B) R') R)) (forall (x:A), ((ex B) ((unique B) (fun (y:B)=> ((R' x) y)))))))))):Prop
% Found relational_choice as proof of a0
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found relational_choice:(forall (A:Type) (B:Type) (R:(A->(B->Prop))), ((forall (x:A), ((ex B) (fun (y:B)=> ((R x) y))))->((ex (A->(B->Prop))) (fun (R':(A->(B->Prop)))=> ((and ((((subrelation A) B) R') R)) (forall (x:A), ((ex B) ((unique B) (fun (y:B)=> ((R' x) y))))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), ((forall (x:A), ((ex B) (fun (y:B)=> ((R x) y))))->((ex (A->(B->Prop))) (fun (R':(A->(B->Prop)))=> ((and ((((subrelation A) B) R') R)) (forall (x:A), ((ex B) ((unique B) (fun (y:B)=> ((R' x) y)))))))))):Prop
% Found relational_choice as proof of b
% Found relational_choice as proof of a0
% 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 eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found eq_sym01:=(eq_sym0 b):(forall (b0:Prop), ((((eq Prop) b) b0)->(((eq Prop) b0) b)))
% Instantiate: b0:=(forall (b0:Prop), ((((eq Prop) b) b0)->(((eq Prop) b0) b))):Prop
% Found eq_sym01 as proof of 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 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 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 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 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 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_sym01:=(eq_sym0 b):(forall (b0:Prop), ((((eq Prop) b) b0)->(((eq Prop) b0) b)))
% Instantiate: b0:=(forall (b0:Prop), ((((eq Prop) b) b0)->(((eq Prop) b0) b))):Prop
% Found eq_sym01 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 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 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: b0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b0
% 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: b0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS 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 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 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 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) 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 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 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 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 conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj 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 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_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: b0:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b0
% Found iff_sym 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 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 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 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) 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 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 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) 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) 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 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 conj as proof of a0
% 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 conj as proof of b0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a0
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a0
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found eq_ref00:=(eq_ref0 a1):(((eq Prop) a1) a1)
% Found (eq_ref0 a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found eq_ref00:=(eq_ref0 a1):(((eq Prop) a1) a1)
% Found (eq_ref0 a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found eq_ref00:=(eq_ref0 a1):(((eq Prop) a1) a1)
% Found (eq_ref0 a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% 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 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% 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) 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 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_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% 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 conj as proof of a0
% 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 conj as proof of b0
% 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) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found eq_ref00:=(eq_ref0 a1):(((eq Prop) a1) a1)
% Found (eq_ref0 a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found eq_ref00:=(eq_ref0 a1):(((eq Prop) a1) a1)
% Found (eq_ref0 a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found eq_ref00:=(eq_ref0 a1):(((eq Prop) a1) a1)
% Found (eq_ref0 a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found ((eq_ref Prop) a1) as proof of (((eq Prop) a1) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) a0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: a1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of a1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: a1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of a1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: a1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of a1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: a1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of a1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: a1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of a1
% 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 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 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% 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 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 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b1:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b1
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b1:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b1
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% 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 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) 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 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) 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 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% 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 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% 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 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) 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 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b00:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b00
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% 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_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_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_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_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) 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 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) 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 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% 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) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% 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) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of a0
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% 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 as proof of a1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found or_ind as proof of a1
% 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 as proof of a1
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: a1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of a1
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) a0)
% Found eq_ref00:=(eq_ref0 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: a1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of a1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: a1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of a1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: a1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of a1
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% 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 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% 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 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 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b1:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b1
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b1:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b1
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b1:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b1
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b1:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b1
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b1
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% 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 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) 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 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% 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 as proof of b00
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b00:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b00
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b00:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b00
% 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 a00):(((eq a) a00) a00)
% Found (eq_ref0 a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% Found ((eq_ref a) a00) as proof of (((eq a) a00) b0)
% 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_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_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b0
% Found classical_choice as proof of a0
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice as proof of a0
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice as proof of a0
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b0
% Found classical_choice as proof of a0
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% 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) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of a0
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of a0
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% 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 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) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% 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 as proof of a1
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found or_ind as proof of a1
% 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 as proof of a1
% 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 as proof of a1
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found ((eq_ref a) b) as proof of (((eq a) b) b00)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 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_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b1:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b1
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b1:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b1
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b1:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b1
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b1
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b1
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b1
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b1
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% 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 as proof of b00
% 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 as proof of b00
% 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_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 or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b00:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b00
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% 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_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_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% 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) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% 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) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b0
% Found classical_choice as proof of a0
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice as proof of a0
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b0
% Found classical_choice as proof of a0
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice as proof of a0
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice as proof of a0
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice as proof of a0
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b0
% Found classical_choice as proof of a0
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b0
% Found classical_choice as proof of a0
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: a0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of a0
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% 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 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% 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 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 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 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% 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: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% 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: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% 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: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% 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) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% 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 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) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found (eq_sym0010 ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx))))) as proof of (P1 b1)
% Found (eq_sym0010 ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx))))) as proof of (P1 b1)
% Found (eq_sym0010 ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx))))) as proof of (P1 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 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 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b1:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b1
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b)
% Found (eq_sym0000 ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx))))) as proof of (P1 b1)
% Found (eq_sym0000 ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx))))) as proof of (P1 b1)
% Found (eq_sym0000 ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx))))) as proof of (P1 b1)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b1:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b1
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b1:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b1
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b1:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b1
% 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) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) Xx)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b0)
% Found (eq_sym0010 ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx))))) as proof of (P1 b1)
% Found (eq_sym0010 ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx))))) as proof of (P1 b1)
% Found (eq_sym0010 ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx))))) as proof of (P1 b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b)
% Found (eq_sym0000 ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx))))) as proof of (P1 b1)
% Found (eq_sym0000 ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx))))) as proof of (P1 b1)
% Found (eq_sym0000 ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx))))) as proof of (P1 b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% 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 as proof of b00
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% 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_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_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_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_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% 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_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_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_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_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found eq_sym000:=(eq_sym00 b):((((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b)->(((eq Prop) b) (forall (Xx:a), ((cT Xx)->(cS Xx)))))
% Found eq_sym000 as proof of (P1 b1)
% Found eq_sym000 as proof of (P1 b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_sym0000:=(fun (x0:(((eq Prop) a0) b0))=> ((eq_sym000 x0) P0)):((((eq Prop) a0) b0)->((P0 b0)->(P0 a0)))
% Found eq_sym0000 as proof of (P1 b1)
% Found eq_sym0000 as proof of (P1 b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% 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) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% 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) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% 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) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% 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) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) a1)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: a0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of a0
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: a0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of a0
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b0
% Found classical_choice as proof of a0
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice as proof of a0
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b
% Found classical_choice as proof of a0
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b0
% Found classical_choice as proof of a0
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found (eq_sym010 ((eq_ref Prop) b)) as proof of a0
% Found (eq_sym010 ((eq_ref Prop) b)) as proof of a0
% Found (eq_sym010 ((eq_ref Prop) b)) as proof of a0
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% 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 a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% 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 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))):(((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) (forall (Xx:a), ((cT Xx)->(cS Xx))))
% Found (eq_ref0 (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) b1)
% Found ((eq_ref Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) as proof of (((eq Prop) (forall (Xx:a), ((cT Xx)->(cS Xx)))) 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 as proof of a0
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 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 as proof of a0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found proj1 as proof of 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% 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: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% 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: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found eq_ref00:=(eq_ref0 b0):(((eq a) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found ((eq_ref a) b0) as proof of (((eq a) b0) b1)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% 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: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% 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: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: a0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of a0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: a0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of a0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% 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: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% 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: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: a0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of a0
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: a0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of a0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: a0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of a0
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b1:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b1
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: a0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of a0
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: a0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of a0
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: a0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of a0
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% 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 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_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b0:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b0
% Found eq_substitution as proof of b1
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% 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 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 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 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 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) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eq_substitution as proof of b1
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) 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 as proof of a0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found proj1 as proof of a0
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b1:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b1
% Found proj1 as proof of a0
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found proj1 as proof of a0
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 proj1:(forall (A:Prop) (B:Prop), (((and A) B)->A))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((and A) B)->A)):Prop
% Found proj1 as proof of b
% Found proj1 as proof of a0
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) 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 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b1)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) 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 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_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b1:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b1
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b1:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b1
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 Xx):(((eq a) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found ((eq_ref a) Xx) as proof of (((eq a) Xx) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq a) b1) b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq a) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found ((eq_ref a) a0) as proof of (((eq a) a0) b1)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b1:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b1
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b1:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b1
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b1:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b1
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b1:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b1
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b1:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b1
% Found eq_ref00:=(eq_ref0 b1):(((eq a) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq a) b1) b0)
% Found ((eq_ref a) b1) as proof of (((eq
% EOF
%------------------------------------------------------------------------------