TSTP Solution File: SEU914^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEU914^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n108.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 300.03s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEU914^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n108.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 11:42:31 CDT 2014
% % CPUTime : 300.03
% 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 0x1df6c20>, <kernel.Type object at 0x1df68c0>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x1df6680>, <kernel.Type object at 0x204fb00>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x1df63b0>, <kernel.DependentProduct object at 0x1df7440>) of role type named cB
% Using role type
% Declaring cB:((b->Prop)->Prop)
% FOF formula (<kernel.Constant object at 0x1df6c20>, <kernel.DependentProduct object at 0x1df7560>) of role type named cA
% Using role type
% Declaring cA:((a->Prop)->Prop)
% FOF formula ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))->(cB (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))) (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:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))))))))))))) of role conjecture named cDOMTHM11_pme
% Conjecture to prove = ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))->(cB (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))) (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:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))))))))))))):Prop
% Parameter b_DUMMY:b.
% Parameter a_DUMMY:a.
% We need to prove ['((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))->(cB (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))) (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:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))))))']
% Parameter b:Type.
% Parameter a:Type.
% Parameter cB:((b->Prop)->Prop).
% Parameter cA:((a->Prop)->Prop).
% Trying to prove ((and (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy0:a)=> False))) (forall (Xx1:(a->Prop)), ((X Xx1)->(forall (Xt:a), ((Xd Xt)->(X (fun (Xz:a)=> ((or (Xx1 Xz)) (((eq a) Xt) Xz)))))))))->(X Xd)))) (forall (Xx1:a), ((Xd Xx1)->(Xx0 Xx1))))) (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy0:b)=> False))) (forall (Xx1:(b->Prop)), ((X Xx1)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx1 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe))))) (forall (Xx1:b), ((Xe Xx1)->(Xy Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe) Xv)))))))))))))))))->(cB (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))) (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:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))))))
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% Found ((eq_ref Prop) (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:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))))))))))))) as proof of (((eq Prop) (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:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))))))))))))) b0)
% Found ((eq_ref Prop) (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:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))) (forall (Xx:(((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 (Xx 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)=> 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Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq 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(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))))))))))))) as proof of (((eq Prop) (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:(((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 (Xx 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 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Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))))))))))))) as proof of (((eq Prop) (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:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and ((and ((and 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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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))))))))))))) b0)
% Found x1:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found (fun (x2:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x1) as proof of ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found (fun (x1:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x2:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x1) as proof of ((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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))))))))))))))))))
% Found (fun (x1:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x2:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x1) as proof of (((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))
% Found (and_rect00 (fun (x1:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x2:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x1)) as proof of ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found ((and_rect0 ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (fun (x1:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x2:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x1)) as proof of ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found (((fun (P:Type) (x1:(((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))) P) x1) x0)) ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (fun (x1:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x2:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x1)) as proof of ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found (fun (x0:((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex 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(((eq (b->Prop)) Xe0) Xv)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))) P) x1) x0)) ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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 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(forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))))))))))
% Found (eq_ref0 (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex 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((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex 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((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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun 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(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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))):(((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) (fun (x:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) as proof of (((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) b0)
% Found ((eta_expansion_dep0 (fun (x1:b)=> Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) as proof of (((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) as proof of (((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) as proof of (((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) b0)
% Found (((eta_expansion_dep b) (fun (x1:b)=> Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) as proof of (((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) b0)
% Found x3:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found (fun (x4:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x3) as proof of ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found (fun (x3:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x4:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x3) as proof of ((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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))))))))))))))))))
% Found (fun (x3:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x4:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x3) as proof of (((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))
% Found (and_rect10 (fun (x3:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x4:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x3)) as proof of ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found ((and_rect1 ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (fun (x3:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x4:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x3)) as proof of ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found (((fun (P:Type) (x3:(((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))) P) x3) x2)) ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (fun (x3:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x4:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x3)) as proof of ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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 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% Found (fun (x2:((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex 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(((eq (b->Prop)) Xe0) Xv)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))) P) x3) x2)) ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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 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% Found (fun (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)) (x2:((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun 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(((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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 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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))))))))))))))))))
% Found (fun (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)) (x2:((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))=> (((fun (P:Type) (x3:(((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->P)))=> (((((and_rect ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))) P) x3) x2)) ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (fun (x3:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x4:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x3))) as proof of (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x0)))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x0)))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x0)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x0)))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x0)))))
% Found (fun (x0:b)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:b), (((eq Prop) (f x)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x))))))
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b0:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))):(((eq Prop) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) 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% Found (eq_ref0 (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex 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((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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun 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(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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex 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((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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (((a->Prop)->((b->Prop)->Prop))->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex 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% Found (eta_expansion00 (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) as proof of (((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) b0)
% Found ((eta_expansion0 Prop) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) as proof of (((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) as proof of (((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) as proof of (((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) b0)
% Found (((eta_expansion b) Prop) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) as proof of (((eq (b->Prop)) (fun (Xx0:b)=> ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))) b0)
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% Found (((eta_expansion (((a->Prop)->((b->Prop)->Prop))->Prop)) Prop) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1:a)=> False))) (forall (Xx1:(a->Prop)), ((X 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% Found (((eta_expansion (((a->Prop)->((b->Prop)->Prop))->Prop)) Prop) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1:a)=> False))) (forall (Xx1:(a->Prop)), ((X 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% Found (((eta_expansion (((a->Prop)->((b->Prop)->Prop))->Prop)) Prop) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1:a)=> False))) (forall (Xx1:(a->Prop)), ((X 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(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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))))))) as proof of (((eq ((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)) (fun (Xe0:(((a->Prop)->((b->Prop)->Prop))->Prop))=> ((and ((and (forall (X:((((a->Prop)->((b->Prop)->Prop))->Prop)->Prop)), (((and (X (fun (Xy:((a->Prop)->((b->Prop)->Prop)))=> False))) (forall (Xx0:(((a->Prop)->((b->Prop)->Prop))->Prop)), ((X Xx0)->(forall (Xt:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xt)->(X (fun (Xz:((a->Prop)->((b->Prop)->Prop)))=> ((or (Xx0 Xz)) (((eq ((a->Prop)->((b->Prop)->Prop))) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(((a->Prop)->((b->Prop)->Prop))->Prop)), (((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:((a->Prop)->((b->Prop)->Prop))), ((Xe0 Xx0)->(Xy Xx0))))->((and ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy0:(b->Prop))=> ((and (cB Xy0)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xy Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (Xe1:(b->Prop))=> ((and ((and ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy1: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 (Xy1: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)->(Xy0 Xx1))))) (forall (Xu:(a->Prop)) (Xv:(b->Prop)), ((iff ((Xr Xu) Xv)) ((and (((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe1) Xv)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xy Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))))))) b0)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x2)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x2)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x2)))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x2)))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x2)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x2)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x2)))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x2)))))
% Found (fun (x2:b)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:b), (((eq Prop) (f x)) ((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S x))))))
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b0
% Found x0:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found (fun (x1:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x0) as proof of ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found (fun (x0:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x1:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x0) as proof of ((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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))))))))))))))))))
% Found (fun (x0:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x1:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x0) as proof of (((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))
% Found (and_rect00 (fun (x0:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x1:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x0)) as proof of ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found ((and_rect0 ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (fun (x0:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x1:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x0)) as proof of ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))) P0) x0) x3)) ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (fun (x0:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x1:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x0)) as proof of ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))
% Found (fun (x3:((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex 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(((eq (b->Prop)) Xe0) Xv)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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), 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(((eq (a->Prop)) Xd) Xu)) (((eq (b->Prop)) Xe0) Xv)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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 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(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)))))))))))))))))) (fun (x0:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x1:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x0))) as proof of (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))
% Found ((conj10 (fun (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)) (x3:((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))=> (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))) P0) x0) x3)) ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (fun (x0:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x1:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x0)))) iff_refl) as proof of (P b0)
% Found (((conj1 b0) (fun (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)) (x3:((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))=> (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))) P0) x0) x3)) ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (fun (x0:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x1:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x0)))) iff_refl) as proof of (P b0)
% Found ((((conj (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))) b0) (fun (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)) (x3:((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))=> (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))->((forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))->P0)))=> (((((and_rect ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))) P0) x0) x3)) ((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (fun (x0:((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))) (x1:(forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))=> x0)))) iff_refl) as proof of (P b0)
% Found ((((conj (forall (Xx:(((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0)))))))->((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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)))))))))))))))))))) b0) (fun (Xx:(((a->Prop)->((b->Prop)->Prop))->Prop)) (x3:((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 (Xx 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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and ((ex ((a->Prop)->((b->Prop)->Prop))) (fun (Xr:((a->Prop)->((b->Prop)->Prop)))=> ((and (Xx Xr)) ((Xr (fun (Xx1:a)=> False)) S))))) (S Xx0))))))))=> (((fun (P0:Type) (x0:(((ex (a->Prop)) (fun (Xx0:(a->Prop))=> ((and (cA Xx0)) ((ex (b->Prop)) (fun (Xy:(b->Prop))=> ((and (cB Xy)) (forall (Xr:((a->Prop)->((b->Prop)->Prop))), ((iff (Xx Xr)) ((ex (a->Prop)) (fun (Xd:(a->Prop))=> ((ex (b->Prop)) (fun (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 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(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)))))))))))))))))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(
% EOF
%------------------------------------------------------------------------------