TSTP Solution File: SEU916^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU916^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n190.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 13:33:22 EDT 2014
% Result : Timeout 300.04s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU916^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n190.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 11:42:41 CDT 2014
% % CPUTime : 300.04
% 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 0x1053830>, <kernel.Type object at 0x1053f38>) of role type named c_type
% Using role type
% Declaring c:Type
% FOF formula (<kernel.Constant object at 0xf3c248>, <kernel.Type object at 0x1053998>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x1053dd0>, <kernel.Type object at 0x1053ef0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1053f38>, <kernel.DependentProduct object at 0x1053680>) of role type named g
% Using role type
% Declaring g:((c->Prop)->(b->Prop))
% FOF formula (<kernel.Constant object at 0x1053998>, <kernel.DependentProduct object at 0x1053d40>) of role type named f
% Using role type
% Declaring f:((c->Prop)->(a->Prop))
% FOF formula (<kernel.Constant object at 0x1053638>, <kernel.DependentProduct object at 0x1053dd0>) of role type named cC
% Using role type
% Declaring cC:((c->Prop)->Prop)
% FOF formula (<kernel.Constant object at 0x1053518>, <kernel.DependentProduct object at 0x1053d40>) of role type named cB
% Using role type
% Declaring cB:((b->Prop)->Prop)
% FOF formula (<kernel.Constant object at 0x1053560>, <kernel.DependentProduct object at 0x1053dd0>) of role type named cA
% Using role type
% Declaring cA:((a->Prop)->Prop)
% FOF formula (((and ((and ((and (forall (Xx:(c->Prop)), ((cC Xx)->(cA (f Xx))))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:a), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->(cB (g Xx)))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:b), ((Xe Xx0)->((g Xy) Xx0)))))))))))))))->((and (forall (Xx:(c->Prop)), ((cC 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 ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((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)->((f Xx) 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)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))) ((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)))))))))))))))))))) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1)))))))))))))))))))))))) of role conjecture named cDOMTHM12_pme
% Conjecture to prove = (((and ((and ((and (forall (Xx:(c->Prop)), ((cC Xx)->(cA (f Xx))))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:a), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->(cB (g Xx)))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:b), ((Xe Xx0)->((g Xy) Xx0)))))))))))))))->((and (forall (Xx:(c->Prop)), ((cC 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 ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((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)->((f Xx) 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)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))) ((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)))))))))))))))))))) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1)))))))))))))))))))))))):Prop
% Parameter c_DUMMY:c.
% Parameter b_DUMMY:b.
% Parameter a_DUMMY:a.
% We need to prove ['(((and ((and ((and (forall (Xx:(c->Prop)), ((cC Xx)->(cA (f Xx))))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:a), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->(cB (g Xx)))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:b), ((Xe Xx0)->((g Xy) Xx0)))))))))))))))->((and (forall (Xx:(c->Prop)), ((cC 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 ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> (((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)->((f Xx) 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)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe)))))))))) ((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)))))))))))))))))))) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))))']
% Parameter c:Type.
% Parameter b:Type.
% Parameter a:Type.
% Parameter g:((c->Prop)->(b->Prop)).
% Parameter f:((c->Prop)->(a->Prop)).
% Parameter cC:((c->Prop)->Prop).
% Parameter cB:((b->Prop)->Prop).
% Parameter cA:((a->Prop)->Prop).
% Trying to prove (((and ((and ((and (forall (Xx:(c->Prop)), ((cC Xx)->(cA (f Xx))))) (forall (Xe:(a->Prop)), (((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx:(a->Prop)), ((X Xx)->(forall (Xt:a), ((Xe Xt)->(X (fun (Xz:a)=> ((or (Xx Xz)) (((eq a) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:a), ((Xe Xx)->((ex (a->Prop)) (fun (S:(a->Prop))=> ((and (cA S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:a), ((Xe Xx0)->((f Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:a), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(c->Prop)), ((cC Xx)->(cB (g Xx)))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:b), ((Xe Xx0)->((g Xx) Xx0))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:b), ((Xe Xx0)->((g 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(((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1)))))))))))))))))))))))
% Found (eq_ref0 (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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 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(Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X 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% Found ((eq_ref Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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 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(forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) as proof of (((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall 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(forall (Xx1:a), ((Xd Xx1)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) as proof of (((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) as proof of (((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) b0)
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b0:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))):(((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and 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% Found ((eq_ref Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) as proof of (((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) as proof of (((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) as proof of (((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) b0)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b0
% Found x2:(cC Xx)
% Found (fun (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))=> x2) as proof of (cC Xx)
% Found (fun (x2:(cC Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))=> x2) as proof of ((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0)))))))))))->(cC Xx))
% Found (fun (x2:(cC Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))=> x2) as proof of ((cC Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0)))))))))))->(cC Xx)))
% Found (and_rect00 (fun (x2:(cC Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))=> x2)) as proof of (cC Xx)
% Found ((and_rect0 (cC Xx)) (fun (x2:(cC Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))=> x2)) as proof of (cC Xx)
% Found (((fun (P:Type) (x2:((cC Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0)))))))))))->P)))=> (((((and_rect (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0)))))))))))) P) x2) x1)) (cC Xx)) (fun (x2:(cC Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))=> x2)) as proof of (cC Xx)
% Found (fun (x1:((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0)))))))))))))=> (((fun (P:Type) (x2:((cC Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0)))))))))))->P)))=> (((((and_rect (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0)))))))))))) P) x2) x1)) (cC Xx)) (fun (x2:(cC Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))=> x2))) as proof of (cC Xx)
% Found (fun (Xx:(c->Prop)) (x1:((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0)))))))))))))=> (((fun (P:Type) (x2:((cC Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0)))))))))))->P)))=> (((((and_rect (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0)))))))))))) P) x2) x1)) (cC Xx)) (fun (x2:(cC Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))=> x2))) as proof of (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx))
% Found (fun (Xx:(c->Prop)) (x1:((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0)))))))))))))=> (((fun (P:Type) (x2:((cC Xx)->((forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0)))))))))))->P)))=> (((((and_rect (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0)))))))))))) P) x2) x1)) (cC Xx)) (fun (x2:(cC Xx)) (x3:(forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))=> x2))) as proof of (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))
% Found eq_ref00:=(eq_ref0 (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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 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False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) 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(forall (Xx1:a), ((Xd Xx1)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) as proof of (((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall 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(forall (Xx1:a), ((Xd Xx1)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) as proof of (((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))) (S Xx)))))))->((and (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->(cC Xx)))) (forall (Xx:(c->Prop)), (((and (cC Xx)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> (((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy: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)->((f Xx) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0))))) (forall (Xx1:b), ((Xe0 Xx1)->((g Xx) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe0))))))))))))->((ex (c->Prop)) (fun (Xe0:(c->Prop))=> ((and ((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe0 Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:c), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(c->Prop)), (((and (cC Xy)) (forall (Xx0:c), ((Xe0 Xx0)->(Xy Xx0))))->((and (cC Xy)) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx0)->((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> (((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)->((f Xy) Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe1 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe1))))) (forall (Xx1:b), ((Xe1 Xx1)->((g Xy) Xx1))))->(forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xx0 Xu) Xv)) ((and (((eq (a->Prop)) Xu) Xd)) (((eq (b->Prop)) Xv) Xe1))))))))))))))))))))))) as proof of (((eq Prop) (forall (Xe:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:((a->Prop)->((b->Prop)->Prop))), ((Xe Xx)->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (S:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((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 (S Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(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), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz))
% EOF
%------------------------------------------------------------------------------