%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV184^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n110.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 13:33:51 EDT 2014

% Result   : Timeout 300.06s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV184^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n110.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 08:23:46 CDT 2014
% % CPUTime  : 300.06 
% 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 0x1e21638>, <kernel.Type object at 0x1e21b48>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x229df80>, <kernel.Type object at 0x1e215f0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1e215a8>, <kernel.DependentProduct object at 0x1e21680>) of role type named cB
% Using role type
% Declaring cB:((b->Prop)->Prop)
% FOF formula (<kernel.Constant object at 0x1e21b48>, <kernel.DependentProduct object at 0x1e21638>) of role type named cA
% Using role type
% Declaring cA:((a->Prop)->Prop)
% FOF formula (((and ((and ((and (forall (X:((a->Prop)->Prop)), ((forall (Xx:(a->Prop)), ((X Xx)->(cA Xx)))->(cA (fun (Xx:a)=> (forall (S:(a->Prop)), ((X S)->(S Xx)))))))) (forall (D:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((D Xx)->(cA Xx)))) ((ex (a->Prop)) (fun (Xy:(a->Prop))=> (D Xy))))) (forall (Xy:(a->Prop)) (Xz:(a->Prop)), ((ex (a->Prop)) (fun (Xw:(a->Prop))=> ((and (forall (Xx:a), ((Xy Xx)->(Xw Xx)))) (forall (Xx:a), ((Xz Xx)->(Xw Xx))))))))->(cA (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (D S)) (S Xx)))))))))) (forall (X:((b->Prop)->Prop)), ((forall (Xx:(b->Prop)), ((X Xx)->(cB Xx)))->(cB (fun (Xx:b)=> (forall (S:(b->Prop)), ((X S)->(S Xx))))))))) (forall (D:((b->Prop)->Prop)), (((and ((and (forall (Xx:(b->Prop)), ((D Xx)->(cB Xx)))) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> (D Xy))))) (forall (Xy:(b->Prop)) (Xz:(b->Prop)), ((ex (b->Prop)) (fun (Xw:(b->Prop))=> ((and (forall (Xx:b), ((Xy Xx)->(Xw Xx)))) (forall (Xx:b), ((Xz Xx)->(Xw Xx))))))))->(cB (fun (Xx:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (D S)) (S Xx)))))))))->((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), ((forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X0:((a->Prop)->Prop)), (((and (X0 (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X0 Xx1)->(forall (Xt:a), ((Xd Xt)->(X0 (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X0 Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X0:((b->Prop)->Prop)), (((and (X0 (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X0 Xx1)->(forall (Xt:b), ((Xe Xt)->(X0 (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X0 Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (forall (S:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X S)->(S Xr)))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X0:((a->Prop)->Prop)), (((and (X0 (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X0 Xx0)->(forall (Xt:a), ((Xd Xt)->(X0 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X0 Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X0:((b->Prop)->Prop)), (((and (X0 (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X0 Xx0)->(forall (Xt:b), ((Xe Xt)->(X0 (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X0 Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv))))))))))))))))))))) of role conjecture named cDOMTHM16_pme
% Conjecture to prove = (((and ((and ((and (forall (X:((a->Prop)->Prop)), ((forall (Xx:(a->Prop)), ((X Xx)->(cA Xx)))->(cA (fun (Xx:a)=> (forall (S:(a->Prop)), ((X S)->(S Xx)))))))) (forall (D:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((D Xx)->(cA Xx)))) ((ex (a->Prop)) (fun (Xy:(a->Prop))=> (D Xy))))) (forall (Xy:(a->Prop)) (Xz:(a->Prop)), ((ex (a->Prop)) (fun (Xw:(a->Prop))=> ((and (forall (Xx:a), ((Xy Xx)->(Xw Xx)))) (forall (Xx:a), ((Xz Xx)->(Xw Xx))))))))->(cA (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (D S)) (S Xx)))))))))) (forall (X:((b->Prop)->Prop)), ((forall (Xx:(b->Prop)), ((X Xx)->(cB Xx)))->(cB (fun (Xx:b)=> (forall (S:(b->Prop)), ((X S)->(S Xx))))))))) (forall (D:((b->Prop)->Prop)), (((and ((and (forall (Xx:(b->Prop)), ((D Xx)->(cB Xx)))) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> (D Xy))))) (forall (Xy:(b->Prop)) (Xz:(b->Prop)), ((ex (b->Prop)) (fun (Xw:(b->Prop))=> ((and (forall (Xx:b), ((Xy Xx)->(Xw Xx)))) (forall (Xx:b), ((Xz Xx)->(Xw Xx))))))))->(cB (fun (Xx:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (D S)) (S Xx)))))))))->((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), ((forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X0:((a->Prop)->Prop)), (((and (X0 (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X0 Xx1)->(forall (Xt:a), ((Xd Xt)->(X0 (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X0 Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X0:((b->Prop)->Prop)), (((and (X0 (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X0 Xx1)->(forall (Xt:b), ((Xe Xt)->(X0 (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X0 Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (forall (S:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X S)->(S Xr)))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X0:((a->Prop)->Prop)), (((and (X0 (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X0 Xx0)->(forall (Xt:a), ((Xd Xt)->(X0 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X0 Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X0:((b->Prop)->Prop)), (((and (X0 (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X0 Xx0)->(forall (Xt:b), ((Xe Xt)->(X0 (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X0 Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv))))))))))))))))))))):Prop
% Parameter b_DUMMY:b.
% Parameter a_DUMMY:a.
% We need to prove ['(((and ((and ((and (forall (X:((a->Prop)->Prop)), ((forall (Xx:(a->Prop)), ((X Xx)->(cA Xx)))->(cA (fun (Xx:a)=> (forall (S:(a->Prop)), ((X S)->(S Xx)))))))) (forall (D:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((D Xx)->(cA Xx)))) ((ex (a->Prop)) (fun (Xy:(a->Prop))=> (D Xy))))) (forall (Xy:(a->Prop)) (Xz:(a->Prop)), ((ex (a->Prop)) (fun (Xw:(a->Prop))=> ((and (forall (Xx:a), ((Xy Xx)->(Xw Xx)))) (forall (Xx:a), ((Xz Xx)->(Xw Xx))))))))->(cA (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (D S)) (S Xx)))))))))) (forall (X:((b->Prop)->Prop)), ((forall (Xx:(b->Prop)), ((X Xx)->(cB Xx)))->(cB (fun (Xx:b)=> (forall (S:(b->Prop)), ((X S)->(S Xx))))))))) (forall (D:((b->Prop)->Prop)), (((and ((and (forall (Xx:(b->Prop)), ((D Xx)->(cB Xx)))) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> (D Xy))))) (forall (Xy:(b->Prop)) (Xz:(b->Prop)), ((ex (b->Prop)) (fun (Xw:(b->Prop))=> ((and (forall (Xx:b), ((Xy Xx)->(Xw Xx)))) (forall (Xx:b), ((Xz Xx)->(Xw Xx))))))))->(cB (fun (Xx:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (D S)) (S Xx)))))))))->((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), ((forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X0:((a->Prop)->Prop)), (((and (X0 (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X0 Xx1)->(forall (Xt:a), ((Xd Xt)->(X0 (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X0 Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X0:((b->Prop)->Prop)), (((and (X0 (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X0 Xx1)->(forall (Xt:b), ((Xe Xt)->(X0 (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X0 Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (forall (S:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X S)->(S Xr)))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X0:((a->Prop)->Prop)), (((and (X0 (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X0 Xx0)->(forall (Xt:a), ((Xd Xt)->(X0 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X0 Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X0:((b->Prop)->Prop)), (((and (X0 (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X0 Xx0)->(forall (Xt:b), ((Xe Xt)->(X0 (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X0 Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))))']
% Parameter b:Type.
% Parameter a:Type.
% Parameter cB:((b->Prop)->Prop).
% Parameter cA:((a->Prop)->Prop).
% Trying to prove (((and ((and ((and (forall (X:((a->Prop)->Prop)), ((forall (Xx:(a->Prop)), ((X Xx)->(cA Xx)))->(cA (fun (Xx:a)=> (forall (S:(a->Prop)), ((X S)->(S Xx)))))))) (forall (D:((a->Prop)->Prop)), (((and ((and (forall (Xx:(a->Prop)), ((D Xx)->(cA Xx)))) ((ex (a->Prop)) (fun (Xy:(a->Prop))=> (D Xy))))) (forall (Xy:(a->Prop)) (Xz:(a->Prop)), ((ex (a->Prop)) (fun (Xw:(a->Prop))=> ((and (forall (Xx:a), ((Xy Xx)->(Xw Xx)))) (forall (Xx:a), ((Xz Xx)->(Xw Xx))))))))->(cA (fun (Xx:a)=> ((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (D S)) (S Xx)))))))))) (forall (X:((b->Prop)->Prop)), ((forall (Xx:(b->Prop)), ((X Xx)->(cB Xx)))->(cB (fun (Xx:b)=> (forall (S:(b->Prop)), ((X S)->(S Xx))))))))) (forall (D:((b->Prop)->Prop)), (((and ((and (forall (Xx:(b->Prop)), ((D Xx)->(cB Xx)))) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> (D Xy))))) (forall (Xy:(b->Prop)) (Xz:(b->Prop)), ((ex (b->Prop)) (fun (Xw:(b->Prop))=> ((and (forall (Xx:b), ((Xy Xx)->(Xw Xx)))) (forall (Xx:b), ((Xz Xx)->(Xw Xx))))))))->(cB (fun (Xx:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (D S)) (S Xx)))))))))->((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), ((forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X0:((a->Prop)->Prop)), (((and (X0 (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X0 Xx1)->(forall (Xt:a), ((Xd Xt)->(X0 (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X0 Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X0:((b->Prop)->Prop)), (((and (X0 (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X0 Xx1)->(forall (Xt:b), ((Xe Xt)->(X0 (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X0 Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (forall (S:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X S)->(S Xr)))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X0:((a->Prop)->Prop)), (((and (X0 (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X0 Xx0)->(forall (Xt:a), ((Xd Xt)->(X0 (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X0 Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X0:((b->Prop)->Prop)), (((and (X0 (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X0 Xx0)->(forall (Xt:b), ((Xe Xt)->(X0 (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X0 Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))))
% Found eq_ref00:=(eq_ref0 (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))):(((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv))))))))))))))))))))
% Found (eq_ref0 (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b0:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))):(((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv))))))))))))))))))))
% Found (eq_ref0 (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found x1:(forall (D:((b->Prop)->Prop)), (((and ((and (forall (Xx:(b->Prop)), ((D Xx)->(cB Xx)))) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> (D Xy))))) (forall (Xy:(b->Prop)) (Xz:(b->Prop)), ((ex (b->Prop)) (fun (Xw:(b->Prop))=> ((and (forall (Xx:b), ((Xy Xx)->(Xw Xx)))) (forall (Xx:b), ((Xz Xx)->(Xw Xx))))))))->(cB (fun (Xx:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (D S)) (S Xx))))))))
% Instantiate: b0:=(forall (D:((b->Prop)->Prop)), (((and ((and (forall (Xx:(b->Prop)), ((D Xx)->(cB Xx)))) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> (D Xy))))) (forall (Xy:(b->Prop)) (Xz:(b->Prop)), ((ex (b->Prop)) (fun (Xw:(b->Prop))=> ((and (forall (Xx:b), ((Xy Xx)->(Xw Xx)))) (forall (Xx:b), ((Xz Xx)->(Xw Xx))))))))->(cB (fun (Xx:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (D S)) (S Xx)))))))):Prop
% Found x1 as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))):(((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv))))))))))))))))))))
% Found (eq_ref0 (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found x1:(forall (D:((b->Prop)->Prop)), (((and ((and (forall (Xx:(b->Prop)), ((D Xx)->(cB Xx)))) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> (D Xy))))) (forall (Xy:(b->Prop)) (Xz:(b->Prop)), ((ex (b->Prop)) (fun (Xw:(b->Prop))=> ((and (forall (Xx:b), ((Xy Xx)->(Xw Xx)))) (forall (Xx:b), ((Xz Xx)->(Xw Xx))))))))->(cB (fun (Xx:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (D S)) (S Xx))))))))
% Instantiate: b0:=(forall (D:((b->Prop)->Prop)), (((and ((and (forall (Xx:(b->Prop)), ((D Xx)->(cB Xx)))) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> (D Xy))))) (forall (Xy:(b->Prop)) (Xz:(b->Prop)), ((ex (b->Prop)) (fun (Xw:(b->Prop))=> ((and (forall (Xx:b), ((Xy Xx)->(Xw Xx)))) (forall (Xx:b), ((Xz Xx)->(Xw Xx))))))))->(cB (fun (Xx:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (D S)) (S Xx)))))))):Prop
% Found x1 as proof of b0
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b0:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))):(((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv))))))))))))))))))))
% Found (eq_ref0 (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop))=> (D Xy))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)) (Xz:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xw:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xy Xx)->(Xw Xx)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xz Xx)->(Xw Xx))))))))->((ex (a->Prop)) (fun (Xx:(a->Prop))=> ((and (cA Xx)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff ((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and (D S)) (S Xr))))) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx0:a), ((Xd Xx0)->(Xx Xx0))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx0:b), ((Xe Xx0)->(Xy Xx0))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))))) as proof of (((eq Prop) (forall (D:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((D Xx)->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz))
% EOF
%------------------------------------------------------------------------------