%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU912^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n091.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 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU912^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n091.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 11:42:26 CDT 2014
% % CPUTime  : 300.02 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x238c638>, <kernel.Type object at 0x238cf38>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (<kernel.Constant object at 0x2764518>, <kernel.Type object at 0x238cdd0>) of role type named b_type
% Using role type
% Declaring b:Type
% FOF formula (<kernel.Constant object at 0x238c7e8>, <kernel.Type object at 0x238c440>) of role type named c_type
% Using role type
% Declaring c:Type
% FOF formula (<kernel.Constant object at 0x238cf38>, <kernel.DependentProduct object at 0x238c560>) of role type named f
% Using role type
% Declaring f:((a->Prop)->(b->Prop))
% FOF formula (<kernel.Constant object at 0x238cdd0>, <kernel.DependentProduct object at 0x238c6c8>) of role type named g
% Using role type
% Declaring g:((b->Prop)->(c->Prop))
% FOF formula (<kernel.Constant object at 0x238c3f8>, <kernel.DependentProduct object at 0x238c7e8>) of role type named cA
% Using role type
% Declaring cA:((a->Prop)->Prop)
% FOF formula (<kernel.Constant object at 0x238c7a0>, <kernel.DependentProduct object at 0x238c6c8>) of role type named cC
% Using role type
% Declaring cC:((c->Prop)->Prop)
% FOF formula (<kernel.Constant object at 0x238c758>, <kernel.DependentProduct object at 0x2271680>) of role type named cB
% Using role type
% Declaring cB:((b->Prop)->Prop)
% FOF formula (((and ((and ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx)))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->(cB Xx)))) (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:b), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx0:b), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:c), ((Xe Xx0)->((g Xy) Xx0)))))))))))))))->((and (forall (Xx:(a->Prop)), ((cA Xx)->(cC (g (f Xx)))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))))))) of role conjecture named cDOMTHM9_pme
% Conjecture to prove = (((and ((and ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx)))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->(cB Xx)))) (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:b), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx0:b), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:c), ((Xe Xx0)->((g Xy) Xx0)))))))))))))))->((and (forall (Xx:(a->Prop)), ((cA Xx)->(cC (g (f Xx)))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))))))):Prop
% Parameter a_DUMMY:a.
% Parameter b_DUMMY:b.
% Parameter c_DUMMY:c.
% We need to prove ['(((and ((and ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx)))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->(cB Xx)))) (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:b), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx0:b), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:c), ((Xe Xx0)->((g Xy) Xx0)))))))))))))))->((and (forall (Xx:(a->Prop)), ((cA Xx)->(cC (g (f Xx)))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))))']
% Parameter a:Type.
% Parameter b:Type.
% Parameter c:Type.
% Parameter f:((a->Prop)->(b->Prop)).
% Parameter g:((b->Prop)->(c->Prop)).
% Parameter cA:((a->Prop)->Prop).
% Parameter cC:((c->Prop)->Prop).
% Parameter cB:((b->Prop)->Prop).
% Trying to prove (((and ((and ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx)))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->(cB Xx)))) (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:b), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx0:b), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:c), ((Xe Xx0)->((g Xy) Xx0)))))))))))))))->((and (forall (Xx:(a->Prop)), ((cA Xx)->(cC (g (f Xx)))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))))
% Found eq_ref00:=(eq_ref0 (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))):(((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))))))
% Found (eq_ref0 (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found x400:=(x40 x6):(cB (f Xx))
% Found (x40 x6) as proof of (cB (f Xx))
% Found ((x4 Xx) x6) as proof of (cB (f Xx))
% Found ((x4 Xx) x6) as proof of (cB (f Xx))
% Found (x30 ((x4 Xx) x6)) as proof of (cC (g (f Xx)))
% Found ((x3 (f Xx)) ((x4 Xx) x6)) as proof of (cC (g (f Xx)))
% Found (fun (x6:(cA Xx))=> ((x3 (f Xx)) ((x4 Xx) x6))) as proof of (cC (g (f Xx)))
% Found (fun (Xx:(a->Prop)) (x6:(cA Xx))=> ((x3 (f Xx)) ((x4 Xx) x6))) as proof of ((cA Xx)->(cC (g (f Xx))))
% Found (fun (Xx:(a->Prop)) (x6:(cA Xx))=> ((x3 (f Xx)) ((x4 Xx) x6))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(cC (g (f Xx)))))
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b0:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b0
% Found eq_ref00:=(eq_ref0 (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))):(((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))))))
% Found (eq_ref0 (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b0
% Found x2:(cA Xx)
% Found (fun (x3:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x2) as proof of (cA Xx)
% Found (fun (x2:(cA Xx)) (x3:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x2) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x2:(cA Xx)) (x3:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x2) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect00 (fun (x2:(cA Xx)) (x3:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x2)) as proof of (cA Xx)
% Found ((and_rect0 (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x2)) as proof of (cA Xx)
% Found (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x2) x1)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x2)) as proof of (cA Xx)
% Found (fun (x1:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x2) x1)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x2))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x1:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x2) x1)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x2))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x1:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x2:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x2) x1)) (cA Xx)) (fun (x2:(cA Xx)) (x3:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x2))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found eq_ref00:=(eq_ref0 (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))):(((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))))))
% Found (eq_ref0 (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found classical_choice:=(fun (A:Type) (B:Type) (R:(A->(B->Prop))) (b:B)=> ((fun (C:((forall (x:A), ((ex B) (fun (y:B)=> (((fun (x0:A) (y0:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y0))) x) y))))->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((fun (x0:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x0) z)))->((R x0) y))) x) (f x)))))))=> (C (fun (x:A)=> ((fun (C0:((or ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))))=> ((((((or_ind ((ex B) (fun (z:B)=> ((R x) z)))) (not ((ex B) (fun (z:B)=> ((R x) z))))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) ((((ex_ind B) (fun (z:B)=> ((R x) z))) ((ex B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y))))) (fun (y:B) (H:((R x) y))=> ((((ex_intro B) (fun (y0:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y0)))) y) (fun (_:((ex B) (fun (z:B)=> ((R x) z))))=> H))))) (fun (N:(not ((ex B) (fun (z:B)=> ((R x) z)))))=> ((((ex_intro B) (fun (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))) b) (fun (H:((ex B) (fun (z:B)=> ((R x) z))))=> ((False_rect ((R x) b)) (N H)))))) C0)) (classic ((ex B) (fun (z:B)=> ((R x) z)))))))) (((choice A) B) (fun (x:A) (y:B)=> (((ex B) (fun (z:B)=> ((R x) z)))->((R x) y)))))):(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x))))))))
% Instantiate: b0:=(forall (A:Type) (B:Type) (R:(A->(B->Prop))), (B->((ex (A->B)) (fun (f:(A->B))=> (forall (x:A), (((ex B) (fun (y:B)=> ((R x) y)))->((R x) (f x)))))))):Prop
% Found classical_choice as proof of b0
% Found x500:=(x50 x4):(cB (f Xx))
% Found (x50 x4) as proof of (cB (f Xx))
% Found ((x5 Xx) x4) as proof of (cB (f Xx))
% Found ((x5 Xx) x4) as proof of (cB (f Xx))
% Found (x30 ((x5 Xx) x4)) as proof of (cC (g (f Xx)))
% Found ((x3 (f Xx)) ((x5 Xx) x4)) as proof of (cC (g (f Xx)))
% Found (fun (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x3 (f Xx)) ((x5 Xx) x4))) as proof of (cC (g (f Xx)))
% Found (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x3 (f Xx)) ((x5 Xx) x4))) as proof of ((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->(cC (g (f Xx))))
% Found (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x3 (f Xx)) ((x5 Xx) x4))) as proof of ((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->(cC (g (f Xx)))))
% Found (and_rect20 (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x3 (f Xx)) ((x5 Xx) x4)))) as proof of (cC (g (f Xx)))
% Found ((and_rect2 (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x3 (f Xx)) ((x5 Xx) x4)))) as proof of (cC (g (f Xx)))
% Found (((fun (P:Type) (x5:((forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0))))))))))))))->P)))=> (((((and_rect (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0))))))))))))))) P) x5) x2)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x3 (f Xx)) ((x5 Xx) x4)))) as proof of (cC (g (f Xx)))
% Found (fun (x4:(cA Xx))=> (((fun (P:Type) (x5:((forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0))))))))))))))->P)))=> (((((and_rect (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0))))))))))))))) P) x5) x2)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x3 (f Xx)) ((x5 Xx) x4))))) as proof of (cC (g (f Xx)))
% Found (fun (Xx:(a->Prop)) (x4:(cA Xx))=> (((fun (P:Type) (x5:((forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0))))))))))))))->P)))=> (((((and_rect (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0))))))))))))))) P) x5) x2)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x3 (f Xx)) ((x5 Xx) x4))))) as proof of ((cA Xx)->(cC (g (f Xx))))
% Found (fun (Xx:(a->Prop)) (x4:(cA Xx))=> (((fun (P:Type) (x5:((forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0))))))))))))))->P)))=> (((((and_rect (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0))))))))))))))) P) x5) x2)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x3 (f Xx)) ((x5 Xx) x4))))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(cC (g (f Xx)))))
% Found x6:(cA Xx)
% Found (fun (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6) as proof of (cA Xx)
% Found (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect20 (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6)) as proof of (cA Xx)
% Found ((and_rect2 (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6)) as proof of (cA Xx)
% Found (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6)) as proof of (cA Xx)
% Found (fun (x5:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found x8:(cA Xx)
% Found (fun (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of (cA Xx)
% Found (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect30 (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found ((and_rect3 (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found (fun (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found x8:(cA Xx)
% Found (fun (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of (cA Xx)
% Found (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect30 (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found ((and_rect3 (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found (fun (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found x4:(cA Xx)
% Found (fun (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4) as proof of (cA Xx)
% Found (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect10 (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4)) as proof of (cA Xx)
% Found ((and_rect1 (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4)) as proof of (cA Xx)
% Found (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x4) x3)) (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4)) as proof of (cA Xx)
% Found (fun (x3:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x4) x3)) (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x4) x3)) (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x4) x3)) (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found eq_ref00:=(eq_ref0 (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))):(((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))))))
% Found (eq_ref0 (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) as proof of (((eq Prop) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))))) b0)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found x8:(cA Xx)
% Found (fun (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of (cA Xx)
% Found (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect30 (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found ((and_rect3 (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found (fun (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found x500:=(x50 x2):(cB (f Xx))
% Found (x50 x2) as proof of (cB (f Xx))
% Found ((x5 Xx) x2) as proof of (cB (f Xx))
% Found ((x5 Xx) x2) as proof of (cB (f Xx))
% Found (x40 ((x5 Xx) x2)) as proof of (cC (g (f Xx)))
% Found ((x4 (f Xx)) ((x5 Xx) x2)) as proof of (cC (g (f Xx)))
% Found (fun (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2))) as proof of (cC (g (f Xx)))
% Found (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2))) as proof of ((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->(cC (g (f Xx))))
% Found (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2))) as proof of ((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->(cC (g (f Xx)))))
% Found (and_rect20 (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2)))) as proof of (cC (g (f Xx)))
% Found ((and_rect2 (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2)))) as proof of (cC (g (f Xx)))
% Found (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2)))) as proof of (cC (g (f Xx)))
% Found (fun (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2))))) as proof of (cC (g (f Xx)))
% Found (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2))))) as proof of ((forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))->(cC (g (f Xx))))
% Found (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2))))) as proof of (((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))->((forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))->(cC (g (f Xx)))))
% Found (and_rect10 (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2)))))) as proof of (cC (g (f Xx)))
% Found ((and_rect1 (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2)))))) as proof of (cC (g (f Xx)))
% Found (((fun (P:Type) (x3:(((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))->((forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx))))) P) x3) x0)) (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2)))))) as proof of (cC (g (f Xx)))
% Found (fun (x2:(cA Xx))=> (((fun (P:Type) (x3:(((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))->((forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx))))) P) x3) x0)) (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2))))))) as proof of (cC (g (f Xx)))
% Found (fun (Xx:(a->Prop)) (x2:(cA Xx))=> (((fun (P:Type) (x3:(((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))->((forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx))))) P) x3) x0)) (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2))))))) as proof of ((cA Xx)->(cC (g (f Xx))))
% Found (fun (Xx:(a->Prop)) (x2:(cA Xx))=> (((fun (P:Type) (x3:(((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))->((forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx))))->P)))=> (((((and_rect ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx))))) P) x3) x0)) (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x2))))))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(cC (g (f Xx)))))
% Found x6:(cA Xx)
% Found (fun (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6) as proof of (cA Xx)
% Found (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect20 (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6)) as proof of (cA Xx)
% Found ((and_rect2 (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6)) as proof of (cA Xx)
% Found (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6)) as proof of (cA Xx)
% Found (fun (x5:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found x6:(cA Xx)
% Found (fun (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6) as proof of (cA Xx)
% Found (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect20 (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6)) as proof of (cA Xx)
% Found ((and_rect2 (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6)) as proof of (cA Xx)
% Found (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6)) as proof of (cA Xx)
% Found (fun (x5:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x5:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))):(((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found iff_trans:=(fun (A:Prop) (B:Prop) (C:Prop) (AB:((iff A) B)) (BC:((iff B) C))=> ((((conj (A->C)) (C->A)) (fun (x:A)=> ((((proj1 (B->C)) (C->B)) BC) ((((proj1 (A->B)) (B->A)) AB) x)))) (fun (x:C)=> ((((proj2 (A->B)) (B->A)) AB) ((((proj2 (B->C)) (C->B)) BC) x))))):(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C))))
% Instantiate: b0:=(forall (A:Prop) (B:Prop) (C:Prop), (((iff A) B)->(((iff B) C)->((iff A) C)))):Prop
% Found iff_trans as proof of b0
% Found x8:(cA Xx)
% Found (fun (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of (cA Xx)
% Found (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect30 (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found ((and_rect3 (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found (fun (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))):(((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))):(((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found x8:(cA Xx)
% Found (fun (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of (cA Xx)
% Found (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect30 (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found ((and_rect3 (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found (fun (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found x8:(cA Xx)
% Found (fun (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of (cA Xx)
% Found (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect30 (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found ((and_rect3 (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8)) as proof of (cA Xx)
% Found (fun (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x7:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x8:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x8) x7)) (cA Xx)) (fun (x8:(cA Xx)) (x9:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x8))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))):(((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found eq_ref00:=(eq_ref0 (g (f Xx))):(((eq (c->Prop)) (g (f Xx))) (g (f Xx)))
% Found (eq_ref0 (g (f Xx))) as proof of (((eq (c->Prop)) (g (f Xx))) b0)
% Found ((eq_ref (c->Prop)) (g (f Xx))) as proof of (((eq (c->Prop)) (g (f Xx))) b0)
% Found ((eq_ref (c->Prop)) (g (f Xx))) as proof of (((eq (c->Prop)) (g (f Xx))) b0)
% Found ((eq_ref (c->Prop)) (g (f Xx))) as proof of (((eq (c->Prop)) (g (f Xx))) b0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))):(((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))) (fun (x:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((x Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X x)))) (forall (Xx0:a), ((x Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((x Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))
% Found (eta_expansion_dep00 (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))) b0)
% Found ((eta_expansion_dep0 (fun (x5:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))) b0)
% Found (((eta_expansion_dep (a->Prop)) (fun (x5:(a->Prop))=> Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))) b0)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))):(((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0))))))))))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xy)) Xx0)))))))))))) b0)
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: b0:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of b0
% Found x4:(cA Xx)
% Found (fun (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4) as proof of (cA Xx)
% Found (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect10 (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4)) as proof of (cA Xx)
% Found ((and_rect1 (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4)) as proof of (cA Xx)
% Found (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x4) x3)) (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4)) as proof of (cA Xx)
% Found (fun (x3:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x4) x3)) (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x4) x3)) (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x4) x3)) (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found eq_ref000:=(eq_ref00 x4):((x4 Xx0)->(x4 Xx0))
% Found (eq_ref00 x4) as proof of ((x4 Xx0)->(Xx Xx0))
% Found ((eq_ref0 Xx0) x4) as proof of ((x4 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(Xx Xx0))
% Found (((eq_ref a) Xx0) x4) as proof of ((x4 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of ((x4 Xx0)->(Xx Xx0))
% Found (fun (Xx0:a)=> (((eq_ref a) Xx0) x4)) as proof of (forall (Xx0:a), ((x4 Xx0)->(Xx Xx0)))
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: b0:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of b0
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: b0:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of b0
% Found x3:(cA Xx)
% Found (fun (x4:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x3) as proof of (cA Xx)
% Found (fun (x3:(cA Xx)) (x4:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x3) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x3:(cA Xx)) (x4:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x3) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect10 (fun (x3:(cA Xx)) (x4:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x3)) as proof of (cA Xx)
% Found ((and_rect1 (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x3)) as proof of (cA Xx)
% Found (((fun (P0:Type) (x3:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P0) x3) x6)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x3)) as proof of (cA Xx)
% Found (fun (x6:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P0:Type) (x3:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P0) x3) x6)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x3))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x6:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P0:Type) (x3:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P0) x3) x6)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x3))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x6:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P0:Type) (x3:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P0) x3) x6)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x3))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found ((conj10 (fun (Xx:(a->Prop)) (x6:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P0:Type) (x3:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P0) x3) x6)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x3)))) iff_trans) as proof of (P b0)
% Found (((conj1 b0) (fun (Xx:(a->Prop)) (x6:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P0:Type) (x3:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P0) x3) x6)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x3)))) iff_trans) as proof of (P b0)
% Found ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) b0) (fun (Xx:(a->Prop)) (x6:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P0:Type) (x3:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P0) x3) x6)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x3)))) iff_trans) as proof of (P b0)
% Found ((((conj (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))) b0) (fun (Xx:(a->Prop)) (x6:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P0:Type) (x3:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P0)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P0) x3) x6)) (cA Xx)) (fun (x3:(cA Xx)) (x4:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x3)))) iff_trans) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 a0):(((eq (b->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (b->Prop)) a0) (f Xx))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (f Xx))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (f Xx))
% Found ((eq_ref (b->Prop)) a0) as proof of (((eq (b->Prop)) a0) (f Xx))
% Found x4:(cA Xx)
% Found (fun (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4) as proof of (cA Xx)
% Found (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect10 (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4)) as proof of (cA Xx)
% Found ((and_rect1 (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4)) as proof of (cA Xx)
% Found (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x4) x3)) (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4)) as proof of (cA Xx)
% Found (fun (x3:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x4) x3)) (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4))) as proof of (cA Xx)
% Found (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x4) x3)) (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4))) as proof of (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx))
% Found (fun (Xx:(a->Prop)) (x3:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x4:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x4) x3)) (cA Xx)) (fun (x4:(cA Xx)) (x5:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x4))) as proof of (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))->(cA Xx)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((g (f Xx)) x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((g (f Xx)) x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((g (f Xx)) x1))
% Found (fun (x1:c)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((g (f Xx)) x1))
% Found (fun (x1:c)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:c), (((eq Prop) (f0 x)) ((g (f Xx)) x)))
% Found eq_ref00:=(eq_ref0 (f0 x1)):(((eq Prop) (f0 x1)) (f0 x1))
% Found (eq_ref0 (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((g (f Xx)) x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((g (f Xx)) x1))
% Found ((eq_ref Prop) (f0 x1)) as proof of (((eq Prop) (f0 x1)) ((g (f Xx)) x1))
% Found (fun (x1:c)=> ((eq_ref Prop) (f0 x1))) as proof of (((eq Prop) (f0 x1)) ((g (f Xx)) x1))
% Found (fun (x1:c)=> ((eq_ref Prop) (f0 x1))) as proof of (forall (x:c), (((eq Prop) (f0 x)) ((g (f Xx)) x)))
% Found functional_extensionality_double:=(fun (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))) (x:(forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y))))=> (((((functional_extensionality_dep A) (fun (x2:A)=> (B->C))) f) g) (fun (x0:A)=> (((((functional_extensionality_dep B) (fun (x3:B)=> C)) (f x0)) (g x0)) (x x0))))):(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g)))
% Instantiate: b0:=(forall (A:Type) (B:Type) (C:Type) (f:(A->(B->C))) (g:(A->(B->C))), ((forall (x:A) (y:B), (((eq C) ((f x) y)) ((g x) y)))->(((eq (A->(B->C))) f) g))):Prop
% Found functional_extensionality_double as proof of b0
% Found x500:=(x50 x0):(cB (f Xx))
% Found (x50 x0) as proof of (cB (f Xx))
% Found ((x5 Xx) x0) as proof of (cB (f Xx))
% Found ((x5 Xx) x0) as proof of (cB (f Xx))
% Found (x40 ((x5 Xx) x0)) as proof of (cC (g (f Xx)))
% Found ((x4 (f Xx)) ((x5 Xx) x0)) as proof of (cC (g (f Xx)))
% Found (fun (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0))) as proof of (cC (g (f Xx)))
% Found (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0))) as proof of ((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->(cC (g (f Xx))))
% Found (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0))) as proof of ((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->(cC (g (f Xx)))))
% Found (and_rect20 (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0)))) as proof of (cC (g (f Xx)))
% Found ((and_rect2 (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0)))) as proof of (cC (g (f Xx)))
% Found (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0)))) as proof of (cC (g (f Xx)))
% Found (fun (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0))))) as proof of (cC (g (f Xx)))
% Found (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0))))) as proof of ((forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))->(cC (g (f Xx))))
% Found (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0))))) as proof of (((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))->((forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))->(cC (g (f Xx)))))
% Found (and_rect10 (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0)))))) as proof of (cC (g (f Xx)))
% Found ((and_rect1 (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0)))))) as proof of (cC (g (f Xx)))
% Found (((fun (P:Type) (x3:(((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))->((forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))->P)))=> (((((and_rect ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))) P) x3) x1)) (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0)))))) as proof of (cC (g (f Xx)))
% Found (fun (x2:(forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:c), ((Xe Xx0)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx0)))))))->((and (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->(cB Xx0)))) (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx00:(b->Prop)), ((X Xx00)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx00 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:b), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx00:b), ((Xe0 Xx00)->(Xy Xx00))))->((and (cB Xy)) (forall (Xx00:c), ((Xe Xx00)->((g Xy) Xx00)))))))))))))))=> (((fun (P:Type) (x3:(((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))->((forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))->P)))=> (((((and_rect ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))) P) x3) x1)) (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0))))))) as proof of (cC (g (f Xx)))
% Found (fun (x1:((and ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))) (x2:(forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:c), ((Xe Xx0)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx0)))))))->((and (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->(cB Xx0)))) (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx00:(b->Prop)), ((X Xx00)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx00 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:b), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx00:b), ((Xe0 Xx00)->(Xy Xx00))))->((and (cB Xy)) (forall (Xx00:c), ((Xe Xx00)->((g Xy) Xx00)))))))))))))))=> (((fun (P:Type) (x3:(((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))->((forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))->P)))=> (((((and_rect ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))) P) x3) x1)) (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0))))))) as proof of ((forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:c), ((Xe Xx0)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx0)))))))->((and (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->(cB Xx0)))) (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx00:(b->Prop)), ((X Xx00)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx00 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:b), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx00:b), ((Xe0 Xx00)->(Xy Xx00))))->((and (cB Xy)) (forall (Xx00:c), ((Xe Xx00)->((g Xy) Xx00))))))))))))))->(cC (g (f Xx))))
% Found (fun (x1:((and ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))) (x2:(forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:c), ((Xe Xx0)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx0)))))))->((and (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->(cB Xx0)))) (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx00:(b->Prop)), ((X Xx00)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx00 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:b), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx00:b), ((Xe0 Xx00)->(Xy Xx00))))->((and (cB Xy)) (forall (Xx00:c), ((Xe Xx00)->((g Xy) Xx00)))))))))))))))=> (((fun (P:Type) (x3:(((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))->((forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))->P)))=> (((((and_rect ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))) P) x3) x1)) (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0))))))) as proof of (((and ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))->((forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:c), ((Xe Xx0)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx0)))))))->((and (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->(cB Xx0)))) (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx00:(b->Prop)), ((X Xx00)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx00 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:b), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx00:b), ((Xe0 Xx00)->(Xy Xx00))))->((and (cB Xy)) (forall (Xx00:c), ((Xe Xx00)->((g Xy) Xx00))))))))))))))->(cC (g (f Xx)))))
% Found (and_rect00 (fun (x1:((and ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA 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(forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA 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(((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0)))))))) as proof of (cC (g (f Xx)))
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(forall (Xx00:b), ((Xe0 Xx00)->(Xy Xx00))))->((and (cB Xy)) (forall (Xx00:c), ((Xe Xx00)->((g Xy) Xx00)))))))))))))))=> (((fun (P:Type) (x3:(((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 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((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))) P) x3) x1)) (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and 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Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall 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(Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx00:b), ((Xe0 Xx00)->(Xy Xx00))))->((and (cB Xy)) (forall (Xx00:c), ((Xe Xx00)->((g Xy) Xx00)))))))))))))))=> (((fun (P:Type) (x3:(((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))->((forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))->P)))=> (((((and_rect ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))) P) x3) x1)) (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun 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Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0)))))))) as proof of (cC (g (f Xx)))
% Found (fun (x0:(cA Xx))=> (((fun (P:Type) (x1:(((and ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) 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(Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))) (x2:(forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:c), ((Xe Xx0)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx0)))))))->((and (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->(cB Xx0)))) (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx00:(b->Prop)), ((X Xx00)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx00 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:b), ((Xe0 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Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))->((forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))->P)))=> (((((and_rect ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))) P) x3) x1)) (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0))))))))) as proof of (cC (g (f Xx)))
% Found (fun (Xx:(a->Prop)) (x0:(cA Xx))=> (((fun (P:Type) (x1:(((and ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx)))))->((forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->(cB Xx)))) (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:b), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx0:b), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:c), ((Xe Xx0)->((g Xy) Xx0))))))))))))))->P)))=> (((((and_rect ((and ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx)))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->(cB Xx)))) (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:b), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx0:b), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:c), ((Xe Xx0)->((g Xy) Xx0))))))))))))))) P) x1) x)) (cC (g (f Xx)))) (fun (x1:((and ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))) (x2:(forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:c), ((Xe Xx0)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx0)))))))->((and (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->(cB Xx0)))) (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx00:(b->Prop)), ((X Xx00)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx00 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:b), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx00:b), ((Xe0 Xx00)->(Xy Xx00))))->((and (cB Xy)) (forall (Xx00:c), ((Xe Xx00)->((g Xy) Xx00)))))))))))))))=> (((fun (P:Type) (x3:(((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))->((forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))->P)))=> (((((and_rect ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))) P) x3) x1)) (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0))))))))) as proof of ((cA Xx)->(cC (g (f Xx))))
% Found (fun (Xx:(a->Prop)) (x0:(cA Xx))=> (((fun (P:Type) (x1:(((and ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx)))))->((forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->(cB Xx)))) (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:b), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx0:b), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:c), ((Xe Xx0)->((g Xy) Xx0))))))))))))))->P)))=> (((((and_rect ((and ((and (forall (Xx:(a->Prop)), ((cA Xx)->(cB (f Xx))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx:(b->Prop)), ((X Xx)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:b), ((Xe Xx)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx)))))))->((and (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->(cA Xx)))) (forall (Xx:(a->Prop)), (((and (cA Xx)) (forall (Xx0:b), ((Xe Xx0)->((f Xx) Xx0))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx0:(a->Prop)), ((X Xx0)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx0 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:a), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx0:a), ((Xe0 Xx0)->(Xy Xx0))))->((and (cA Xy)) (forall (Xx0:b), ((Xe Xx0)->((f Xy) Xx0)))))))))))))))) (forall (Xx:(b->Prop)), ((cB Xx)->(cC (g Xx)))))) (forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx:(c->Prop)), ((X Xx)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx:c), ((Xe Xx)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx)))))))->((and (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->(cB Xx)))) (forall (Xx:(b->Prop)), (((and (cB Xx)) (forall (Xx0:c), ((Xe Xx0)->((g Xx) Xx0))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx0:b), ((Xe0 Xx0)->(Xx Xx0))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx0:b), ((Xe0 Xx0)->(Xy Xx0))))->((and (cB Xy)) (forall (Xx0:c), ((Xe Xx0)->((g Xy) Xx0))))))))))))))) P) x1) x)) (cC (g (f Xx)))) (fun (x1:((and ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))) (x2:(forall (Xe:(c->Prop)), (((and (forall (X:((c->Prop)->Prop)), (((and (X (fun (Xy:c)=> False))) (forall (Xx0:(c->Prop)), ((X Xx0)->(forall (Xt:c), ((Xe Xt)->(X (fun (Xz:c)=> ((or (Xx0 Xz)) (((eq c) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:c), ((Xe Xx0)->((ex (c->Prop)) (fun (S:(c->Prop))=> ((and (cC S)) (S Xx0)))))))->((and (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->(cB Xx0)))) (forall (Xx0:(b->Prop)), (((and (cB Xx0)) (forall (Xx00:c), ((Xe Xx00)->((g Xx0) Xx00))))->((ex (b->Prop)) (fun (Xe0:(b->Prop))=> ((and ((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx00:(b->Prop)), ((X Xx00)->(forall (Xt:b), ((Xe0 Xt)->(X (fun (Xz:b)=> ((or (Xx00 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:b), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(b->Prop)), (((and (cB Xy)) (forall (Xx00:b), ((Xe0 Xx00)->(Xy Xx00))))->((and (cB Xy)) (forall (Xx00:c), ((Xe Xx00)->((g Xy) Xx00)))))))))))))))=> (((fun (P:Type) (x3:(((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))->((forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))->P)))=> (((((and_rect ((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0))))) P) x3) x1)) (cC (g (f Xx)))) (fun (x3:((and (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))) (x4:(forall (Xx0:(b->Prop)), ((cB Xx0)->(cC (g Xx0)))))=> (((fun (P:Type) (x5:((forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))->((forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))->P)))=> (((((and_rect (forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00))))))))))))))) P) x5) x3)) (cC (g (f Xx)))) (fun (x5:(forall (Xx0:(a->Prop)), ((cA Xx0)->(cB (f Xx0))))) (x6:(forall (Xe:(b->Prop)), (((and (forall (X:((b->Prop)->Prop)), (((and (X (fun (Xy:b)=> False))) (forall (Xx0:(b->Prop)), ((X Xx0)->(forall (Xt:b), ((Xe Xt)->(X (fun (Xz:b)=> ((or (Xx0 Xz)) (((eq b) Xt) Xz)))))))))->(X Xe)))) (forall (Xx0:b), ((Xe Xx0)->((ex (b->Prop)) (fun (S:(b->Prop))=> ((and (cB S)) (S Xx0)))))))->((and (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->(cA Xx0)))) (forall (Xx0:(a->Prop)), (((and (cA Xx0)) (forall (Xx00:b), ((Xe Xx00)->((f Xx0) Xx00))))->((ex (a->Prop)) (fun (Xe0:(a->Prop))=> ((and ((and (forall (X:((a->Prop)->Prop)), (((and (X (fun (Xy:a)=> False))) (forall (Xx00:(a->Prop)), ((X Xx00)->(forall (Xt:a), ((Xe0 Xt)->(X (fun (Xz:a)=> ((or (Xx00 Xz)) (((eq a) Xt) Xz)))))))))->(X Xe0)))) (forall (Xx00:a), ((Xe0 Xx00)->(Xx0 Xx00))))) (forall (Xy:(a->Prop)), (((and (cA Xy)) (forall (Xx00:a), ((Xe0 Xx00)->(Xy Xx00))))->((and (cA Xy)) (forall (Xx00:b), ((Xe Xx00)->((f Xy) Xx00)))))))))))))))=> ((x4 (f Xx)) ((x5 Xx) x0))))))))) as proof of (forall (Xx:(a->Prop)), ((cA Xx)->(cC (g (f Xx)))))
% Found x6:(cA Xx)
% Found (fun (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6) as proof of (cA Xx)
% Found (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6) as proof of ((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx))
% Found (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6) as proof of ((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->(cA Xx)))
% Found (and_rect20 (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6)) as proof of (cA Xx)
% Found ((and_rect2 (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6)) as proof of (cA Xx)
% Found (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0))))=> x6)) as proof of (cA Xx)
% Found (fun (x5:((and (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))))=> (((fun (P:Type) (x6:((cA Xx)->((forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))->P)))=> (((((and_rect (cA Xx)) (forall (Xx0:c), ((Xe Xx0)->((g (f Xx)) Xx0)))) P) x6) x5)) (cA Xx)) (fun (x6:(cA Xx)) (x7:(forall (Xx0:c), ((Xe Xx0)->(
% EOF
%------------------------------------------------------------------------------