%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV039^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n187.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 13:33:38 EDT 2014

% Result   : Timeout 300.01s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV039^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n187.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 07:40:16 CDT 2014
% % CPUTime  : 300.01 
% 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 0x1230680>, <kernel.Type object at 0x12305f0>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula ((ex ((a->(a->Prop))->((a->Prop)->Prop))) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U)))))))) of role conjecture named cTHM265_pme
% Conjecture to prove = ((ex ((a->(a->Prop))->((a->Prop)->Prop))) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U)))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['((ex ((a->(a->Prop))->((a->Prop)->Prop))) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U))))))))']
% Parameter a:Type.
% Trying to prove ((ex ((a->(a->Prop))->((a->Prop)->Prop))) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U))))))))
% Found eq_ref00:=(eq_ref0 (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U)))))))):(((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U)))))))) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U))))))))
% Found (eq_ref0 (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U)))))))) as proof of (((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U)))))))) b)
% Found ((eq_ref (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U)))))))) as proof of (((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U)))))))) b)
% Found ((eq_ref (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U)))))))) as proof of (((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U)))))))) b)
% Found ((eq_ref (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U)))))))) as proof of (((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U)))))))) b)
% Found x000:=(x00 x0):((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (x00 x0) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (x00 x0) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (x00:((x R) Xp))=> (x00 x0)) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (Xp:(a->Prop)) (x00:((x R) Xp))=> (x00 x0)) as proof of (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))
% Found (fun (Xp:(a->Prop)) (x00:((x R) Xp))=> (x00 x0)) as proof of (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))
% Found eq_ref00:=(eq_ref0 b):(((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) b) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U))))))))
% Found ((eq_ref (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) b) as proof of (((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) b) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U))))))))
% Found ((eq_ref (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) b) as proof of (((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) b) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U))))))))
% Found ((eq_ref (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) b) as proof of (((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) b) (fun (F:((a->(a->Prop))->((a->Prop)->Prop)))=> ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((F R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((F R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((F R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (F S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (F T)) (F U))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) a0) b)
% Found ((eq_ref (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) a0) as proof of (((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) a0) b)
% Found ((eq_ref (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) a0) as proof of (((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) a0) b)
% Found ((eq_ref (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) a0) as proof of (((eq (((a->(a->Prop))->((a->Prop)->Prop))->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))):(((eq Prop) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))
% Found (eq_ref0 (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))) as proof of (((eq Prop) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))) as proof of (((eq Prop) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))) as proof of (((eq Prop) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))) b)
% Found ((eq_ref Prop) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))) as proof of (((eq Prop) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (((eq_trans000 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((((eq_trans00 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (((((eq_trans0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (((eq_trans000 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((((eq_trans00 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (((((eq_trans0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found x01:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))
% Found x01 as proof of (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))
% Found x00:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))
% Found x00 as proof of ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))
% Found x0:((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))
% Found x0 as proof of ((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))
% Found (((x02 x01) x0) x00) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (((x02 x01) x0) x00) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (x02:((x R) Xp))=> (((x02 x01) x0) x00)) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (Xp:(a->Prop)) (x02:((x R) Xp))=> (((x02 x01) x0) x00)) as proof of (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))
% Found (fun (Xp:(a->Prop)) (x02:((x R) Xp))=> (((x02 x01) x0) x00)) as proof of (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))
% Found eq_ref00:=(eq_ref0 (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S)))))))):(((eq Prop) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S)))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))
% Found (eq_ref0 (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S)))))))) as proof of (((eq Prop) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S)))))))) b)
% Found ((eq_ref Prop) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S)))))))) as proof of (((eq Prop) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S)))))))) b)
% Found ((eq_ref Prop) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S)))))))) as proof of (((eq Prop) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S)))))))) b)
% Found ((eq_ref Prop) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S)))))))) as proof of (((eq Prop) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S)))))))) b)
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))):(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))):(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x))
% Found ((((eq_trans00 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x))
% Found (((((eq_trans0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x))
% Found (eq_sym000 ((((((eq_trans Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_sym00 (f x)) ((((((eq_trans Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (((eq_sym0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((((eq_sym Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found x00:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))
% Found x00 as proof of ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))
% Found x01:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))
% Found x01 as proof of (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))
% Found x02:(forall (Xx:a), ((R Xx) Xx))
% Found x02 as proof of (forall (Xx:a), ((R Xx) Xx))
% Found x0:((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))
% Found x0 as proof of ((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))
% Found x03:(forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))
% Found x03 as proof of (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx)))
% Found (((((x04 x03) x01) x02) x00) x0) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (((((x04 x03) x01) x02) x00) x0) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (x04:((x R) Xp))=> (((((x04 x03) x01) x02) x00) x0)) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (Xp:(a->Prop)) (x04:((x R) Xp))=> (((((x04 x03) x01) x02) x00) x0)) as proof of (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))
% Found (fun (Xp:(a->Prop)) (x04:((x R) Xp))=> (((((x04 x03) x01) x02) x00) x0)) as proof of (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))
% Found eq_ref00:=(eq_ref0 (x x00)):(((eq ((a->Prop)->Prop)) (x x00)) (x x00))
% Found (eq_ref0 (x x00)) as proof of (((eq ((a->Prop)->Prop)) (x x00)) P)
% Found ((eq_ref ((a->Prop)->Prop)) (x x00)) as proof of (((eq ((a->Prop)->Prop)) (x x00)) P)
% Found ((eq_ref ((a->Prop)->Prop)) (x x00)) as proof of (((eq ((a->Prop)->Prop)) (x x00)) P)
% Found ((eq_ref ((a->Prop)->Prop)) (x x00)) as proof of (((eq ((a->Prop)->Prop)) (x x00)) P)
% Found (eq_sym000 ((eq_ref ((a->Prop)->Prop)) (x x00))) as proof of (((eq ((a->Prop)->Prop)) P) (x x00))
% Found ((eq_sym00 P) ((eq_ref ((a->Prop)->Prop)) (x x00))) as proof of (((eq ((a->Prop)->Prop)) P) (x x00))
% Found (((eq_sym0 (x x00)) P) ((eq_ref ((a->Prop)->Prop)) (x x00))) as proof of (((eq ((a->Prop)->Prop)) P) (x x00))
% Found ((((eq_sym ((a->Prop)->Prop)) (x x00)) P) ((eq_ref ((a->Prop)->Prop)) (x x00))) as proof of (((eq ((a->Prop)->Prop)) P) (x x00))
% Found ((((eq_sym ((a->Prop)->Prop)) (x x00)) P) ((eq_ref ((a->Prop)->Prop)) (x x00))) as proof of (((eq ((a->Prop)->Prop)) P) (x x00))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))):(((eq Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))
% Found (eq_ref0 (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) as proof of (((eq Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) as proof of (((eq Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) as proof of (((eq Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) as proof of (((eq Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))):(((eq ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S)))))
% Found (eq_ref0 (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found x1:(P Xq)
% Instantiate: x00:=Xq:(a->Prop)
% Found (fun (x1:(P Xq))=> x1) as proof of (P x00)
% Found (fun (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x00)
% Found x1:(P Xq)
% Instantiate: x00:=Xq:(a->Prop)
% Found (fun (x1:(P Xq))=> x1) as proof of (P x00)
% Found (fun (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x00))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found x00:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))
% Instantiate: x:=(fun (x2:(a->(a->Prop))) (x10:(a->Prop))=> (forall (Xx0:a), ((x2 Xx0) Xx0))):((a->(a->Prop))->((a->Prop)->Prop));x02:=(fun (x1:a)=> (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))):(a->Prop)
% Found x00 as proof of ((and ((x R) x02)) (x02 Xx))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x00))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))))
% Found (((eq_trans000 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))))
% Found ((((eq_trans00 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))))
% Found (((((eq_trans0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))))
% Found (((eq_trans000 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))))
% Found ((((eq_trans00 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))))
% Found (((((eq_trans0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))))
% Found x00:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))
% Instantiate: x:=(fun (x2:(a->(a->Prop))) (x10:(a->Prop))=> (forall (Xx0:a), ((x2 Xx0) Xx0))):((a->(a->Prop))->((a->Prop)->Prop));x02:=(fun (x1:a)=> (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))):(a->Prop)
% Found x00 as proof of ((and ((x R) x02)) (x02 Xx))
% Found x01:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))
% Instantiate: x:=(fun (x2:(a->(a->Prop))) (x10:(a->Prop))=> (forall (Xx0:a), ((x2 Xx0) Xx0))):((a->(a->Prop))->((a->Prop)->Prop));x00:=(fun (x1:a)=> (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))):(a->Prop)
% Found x01 as proof of ((and ((x R) x00)) (x00 Xx))
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% 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: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found x01:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))
% Instantiate: x:=(fun (x2:(a->(a->Prop))) (x10:(a->Prop))=> (forall (Xx0:a), ((x2 Xx0) Xx0))):((a->(a->Prop))->((a->Prop)->Prop));x00:=(fun (x1:a)=> (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))):(a->Prop)
% Found (fun (x02:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> x01) as proof of ((and ((x R) x00)) (x00 Xx))
% Found (fun (x01:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x02:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> x01) as proof of ((forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz)))->((and ((x R) x00)) (x00 Xx)))
% Found (fun (x01:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x02:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> x01) as proof of (((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))->((forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz)))->((and ((x R) x00)) (x00 Xx))))
% Found (and_rect00 (fun (x01:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x02:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> x01)) as proof of ((and ((x R) x00)) (x00 Xx))
% Found ((and_rect0 ((and ((x R) x00)) (x00 Xx))) (fun (x01:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x02:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> x01)) as proof of ((and ((x R) x00)) (x00 Xx))
% Found (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x0)) ((and ((x R) x00)) (x00 Xx))) (fun (x01:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x02:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> x01)) as proof of ((and ((x R) x00)) (x00 Xx))
% Found (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x0)) ((and ((x R) x00)) (x00 Xx))) (fun (x01:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))) (x02:(forall (Xx0:a) (Xy:a) (Xz:a), (((and ((R Xx0) Xy)) ((R Xy) Xz))->((R Xx0) Xz))))=> x01)) as proof of ((and ((x R) x00)) (x00 Xx))
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion0 Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x00)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion_dep000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion_dep00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 x00):(((eq (a->Prop)) x00) x00)
% Found (eq_ref0 x00) as proof of (((eq (a->Prop)) x00) Xq)
% Found ((eq_ref (a->Prop)) x00) as proof of (((eq (a->Prop)) x00) Xq)
% Found ((eq_ref (a->Prop)) x00) as proof of (((eq (a->Prop)) x00) Xq)
% Found ((eq_ref (a->Prop)) x00) as proof of (((eq (a->Prop)) x00) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x00)) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x00)) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eq_sym0 x00) Xq) ((eq_ref (a->Prop)) x00)) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((((eq_sym (a->Prop)) x00) Xq) ((eq_ref (a->Prop)) x00)) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> ((((eq_sym (a->Prop)) x00) Xq) ((eq_ref (a->Prop)) x00))) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x01:((x R) Xq)) (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x00))
% Found (fun (x01:((x R) Xq)) (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((x R) Xq)->((Xq Xx)->(((eq (a->Prop)) Xq) x00)))
% Found (and_rect00 (fun (x01:((x R) Xq)) (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((and_rect0 (((eq (a->Prop)) Xq) x00)) (fun (x01:((x R) Xq)) (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((fun (P:Type) (x1:(((x R) Xq)->((Xq Xx)->P)))=> (((((and_rect ((x R) Xq)) (Xq Xx)) P) x1) x000)) (((eq (a->Prop)) Xq) x00)) (fun (x01:((x R) Xq)) (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((fun (P:Type) (x1:(((x R) Xq)->((Xq Xx)->P)))=> (((((and_rect ((x R) Xq)) (Xq Xx)) P) x1) x000)) (((eq (a->Prop)) Xq) x00)) (fun (x01:((x R) Xq)) (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion0 Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x00)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion_dep000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion_dep00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found x1:(P Xq)
% Instantiate: x02:=Xq:(a->Prop)
% Found (fun (x1:(P Xq))=> x1) as proof of (P x02)
% Found (fun (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x02)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x02))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eq_ref00:=(eq_ref0 (x x02)):(((eq ((a->Prop)->Prop)) (x x02)) (x x02))
% Found (eq_ref0 (x x02)) as proof of (((eq ((a->Prop)->Prop)) (x x02)) P)
% Found ((eq_ref ((a->Prop)->Prop)) (x x02)) as proof of (((eq ((a->Prop)->Prop)) (x x02)) P)
% Found ((eq_ref ((a->Prop)->Prop)) (x x02)) as proof of (((eq ((a->Prop)->Prop)) (x x02)) P)
% Found ((eq_ref ((a->Prop)->Prop)) (x x02)) as proof of (((eq ((a->Prop)->Prop)) (x x02)) P)
% Found (eq_sym000 ((eq_ref ((a->Prop)->Prop)) (x x02))) as proof of (((eq ((a->Prop)->Prop)) P) (x x02))
% Found ((eq_sym00 P) ((eq_ref ((a->Prop)->Prop)) (x x02))) as proof of (((eq ((a->Prop)->Prop)) P) (x x02))
% Found (((eq_sym0 (x x02)) P) ((eq_ref ((a->Prop)->Prop)) (x x02))) as proof of (((eq ((a->Prop)->Prop)) P) (x x02))
% Found ((((eq_sym ((a->Prop)->Prop)) (x x02)) P) ((eq_ref ((a->Prop)->Prop)) (x x02))) as proof of (((eq ((a->Prop)->Prop)) P) (x x02))
% Found ((((eq_sym ((a->Prop)->Prop)) (x x02)) P) ((eq_ref ((a->Prop)->Prop)) (x x02))) as proof of (((eq ((a->Prop)->Prop)) P) (x x02))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found x1:(P Xq)
% Instantiate: x02:=Xq:(a->Prop)
% Found (fun (x1:(P Xq))=> x1) as proof of (P x02)
% Found (fun (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x02)
% Found eq_ref00:=(eq_ref0 (x x00)):(((eq ((a->Prop)->Prop)) (x x00)) (x x00))
% Found (eq_ref0 (x x00)) as proof of (((eq ((a->Prop)->Prop)) (x x00)) P)
% Found ((eq_ref ((a->Prop)->Prop)) (x x00)) as proof of (((eq ((a->Prop)->Prop)) (x x00)) P)
% Found ((eq_ref ((a->Prop)->Prop)) (x x00)) as proof of (((eq ((a->Prop)->Prop)) (x x00)) P)
% Found ((eq_ref ((a->Prop)->Prop)) (x x00)) as proof of (((eq ((a->Prop)->Prop)) (x x00)) P)
% Found (eq_sym000 ((eq_ref ((a->Prop)->Prop)) (x x00))) as proof of (((eq ((a->Prop)->Prop)) P) (x x00))
% Found ((eq_sym00 P) ((eq_ref ((a->Prop)->Prop)) (x x00))) as proof of (((eq ((a->Prop)->Prop)) P) (x x00))
% Found (((eq_sym0 (x x00)) P) ((eq_ref ((a->Prop)->Prop)) (x x00))) as proof of (((eq ((a->Prop)->Prop)) P) (x x00))
% Found ((((eq_sym ((a->Prop)->Prop)) (x x00)) P) ((eq_ref ((a->Prop)->Prop)) (x x00))) as proof of (((eq ((a->Prop)->Prop)) P) (x x00))
% Found ((((eq_sym ((a->Prop)->Prop)) (x x00)) P) ((eq_ref ((a->Prop)->Prop)) (x x00))) as proof of (((eq ((a->Prop)->Prop)) P) (x x00))
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x02))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->(((eq (a->Prop)) Xq) x00))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq)) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->(((eq (a->Prop)) Xq) x00)))
% Found (and_rect00 (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((and_rect0 (((eq (a->Prop)) Xq) x00)) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x0)) (((eq (a->Prop)) Xq) x00)) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x0)) (((eq (a->Prop)) Xq) x00)) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))):(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))):(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))):(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))):(((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found (eq_ref0 (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x00 x)))
% Found (functional_extensionality0000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((functional_extensionality000 x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((functional_extensionality00 Xq) x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((((functional_extensionality0 Prop) Xq) x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((((functional_extensionality a) Prop) Xq) x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x00 x)))
% Found (functional_extensionality_dep0000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((functional_extensionality_dep000 x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((functional_extensionality_dep00 Xq) x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((((functional_extensionality_dep0 (fun (x3:a)=> Prop)) Xq) x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((((functional_extensionality_dep a) (fun (x3:a)=> Prop)) Xq) x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))):(((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))
% Found (eq_ref0 (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found ((eq_ref ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) as proof of (((eq ((a->Prop)->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))):(((eq Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))
% Found (eq_ref0 (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) as proof of (((eq Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) as proof of (((eq Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) as proof of (((eq Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) as proof of (((eq Prop) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp)))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))):(((eq ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S)))))
% Found (eq_ref0 (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))) b)
% Found x1:(P Xq)
% Instantiate: x00:=Xq:(a->Prop)
% Found (fun (x1:(P Xq))=> x1) as proof of (P x00)
% Found (fun (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x00))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found x020:=(x02 x0):((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (x02 x0) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (x02 x0) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (x02:((x R) Xp))=> (x02 x0)) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (Xp:(a->Prop)) (x02:((x R) Xp))=> (x02 x0)) as proof of (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))
% Found (fun (Xp:(a->Prop)) (x02:((x R) Xp))=> (x02 x0)) as proof of (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))
% Found ((conj20 (fun (Xp:(a->Prop)) (x02:((x R) Xp))=> (x02 x0))) eq_sym) as proof of (P b)
% Found (((conj2 b) (fun (Xp:(a->Prop)) (x02:((x R) Xp))=> (x02 x0))) eq_sym) as proof of (P b)
% Found ((((conj (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) b) (fun (Xp:(a->Prop)) (x02:((x R) Xp))=> (x02 x0))) eq_sym) as proof of (P b)
% Found ((((conj (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) b) (fun (Xp:(a->Prop)) (x02:((x R) Xp))=> (x02 x0))) eq_sym) as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found x1:(P Xq)
% Instantiate: x02:=Xq:(a->Prop)
% Found (fun (x1:(P Xq))=> x1) as proof of (P x02)
% Found (fun (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion0 Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x00)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion_dep000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion_dep00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found x1:(P Xq)
% Instantiate: x02:=Xq:(a->Prop)
% Found (fun (x1:(P Xq))=> x1) as proof of (P x02)
% Found (fun (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x02)
% Found eq_ref00:=(eq_ref0 x00):(((eq (a->Prop)) x00) x00)
% Found (eq_ref0 x00) as proof of (((eq (a->Prop)) x00) Xq)
% Found ((eq_ref (a->Prop)) x00) as proof of (((eq (a->Prop)) x00) Xq)
% Found ((eq_ref (a->Prop)) x00) as proof of (((eq (a->Prop)) x00) Xq)
% Found ((eq_ref (a->Prop)) x00) as proof of (((eq (a->Prop)) x00) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x00)) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x00)) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eq_sym0 x00) Xq) ((eq_ref (a->Prop)) x00)) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((((eq_sym (a->Prop)) x00) Xq) ((eq_ref (a->Prop)) x00)) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> ((((eq_sym (a->Prop)) x00) Xq) ((eq_ref (a->Prop)) x00))) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found x00:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))
% Instantiate: x:=(fun (x2:(a->(a->Prop))) (x10:(a->Prop))=> (forall (Xx0:a), ((x2 Xx0) Xx0))):((a->(a->Prop))->((a->Prop)->Prop));x04:=(fun (x1:a)=> (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))):(a->Prop)
% Found x00 as proof of ((and ((x R) x04)) (x04 Xx))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found (fun (x00:(a->Prop))=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found (fun (x00:(a->Prop))=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:(a->Prop)), (((eq Prop) (f x0)) ((and ((and ((x R) x0)) (x0 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found (fun (x00:(a->Prop))=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found (fun (x00:(a->Prop))=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:(a->Prop)), (((eq Prop) (f x0)) ((and ((and ((x R) x0)) (x0 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x02))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found x1:(P Xq)
% Instantiate: x00:=Xq:(a->Prop)
% Found (fun (x1:(P Xq))=> x1) as proof of (P x00)
% Found (fun (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x00)
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))):(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))):(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (eq_ref0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) b)
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found (fun (x00:(a->Prop))=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found (fun (x00:(a->Prop))=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:(a->Prop)), (((eq Prop) (f x0)) ((and ((and ((x R) x0)) (x0 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found eq_ref00:=(eq_ref0 (f x00)):(((eq Prop) (f x00)) (f x00))
% Found (eq_ref0 (f x00)) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found ((eq_ref Prop) (f x00)) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found (fun (x00:(a->Prop))=> ((eq_ref Prop) (f x00))) as proof of (((eq Prop) (f x00)) ((and ((and ((x R) x00)) (x00 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x00)))))
% Found (fun (x00:(a->Prop))=> ((eq_ref Prop) (f x00))) as proof of (forall (x0:(a->Prop)), (((eq Prop) (f x0)) ((and ((and ((x R) x0)) (x0 Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) x0))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))->(P (f x))))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))->(P (f x))))
% Found ((((eq_trans00 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))->(P (f x))))
% Found (((((eq_trans0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))->(P (f x))))
% Found ((((((eq_trans Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))->(P (f x))))
% Found ((((((eq_trans Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))->(P (f x))))
% Found ((((((eq_trans Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x))
% Found (eq_sym000 ((((((eq_trans Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_sym00 (f x)) ((((((eq_trans Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (((eq_sym0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (((eq_sym0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((((eq_sym Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x01:((x R) Xq)) (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of ((Xq Xx)->(((eq (a->Prop)) Xq) x00))
% Found (fun (x01:((x R) Xq)) (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)) as proof of (((x R) Xq)->((Xq Xx)->(((eq (a->Prop)) Xq) x00)))
% Found (and_rect00 (fun (x01:((x R) Xq)) (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((and_rect0 (((eq (a->Prop)) Xq) x00)) (fun (x01:((x R) Xq)) (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((fun (P:Type) (x1:(((x R) Xq)->((Xq Xx)->P)))=> (((((and_rect ((x R) Xq)) (Xq Xx)) P) x1) x000)) (((eq (a->Prop)) Xq) x00)) (fun (x01:((x R) Xq)) (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((fun (P:Type) (x1:(((x R) Xq)->((Xq Xx)->P)))=> (((((and_rect ((x R) Xq)) (Xq Xx)) P) x1) x000)) (((eq (a->Prop)) Xq) x00)) (fun (x01:((x R) Xq)) (x02:(Xq Xx))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found x020:=(x02 x0):((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (x02 x0) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (x02 x0) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (x02:((x R) Xp))=> (x02 x0)) as proof of ((ex a) (fun (Xz:a)=> (Xp Xz)))
% Found (fun (Xp:(a->Prop)) (x02:((x R) Xp))=> (x02 x0)) as proof of (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz))))
% Found (fun (Xp:(a->Prop)) (x02:((x R) Xp))=> (x02 x0)) as proof of (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion_dep000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion_dep00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x00)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion0 Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found x00:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))
% Instantiate: x:=(fun (x2:(a->(a->Prop))) (x10:(a->Prop))=> (forall (Xx0:a), ((x2 Xx0) Xx0))):((a->(a->Prop))->((a->Prop)->Prop));x04:=(fun (x1:a)=> (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))):(a->Prop)
% Found x00 as proof of ((and ((x R) x04)) (x04 Xx))
% Found x00:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))
% Instantiate: x:=(fun (x2:(a->(a->Prop))) (x10:(a->Prop))=> (forall (Xx0:a), ((x2 Xx0) Xx0))):((a->(a->Prop))->((a->Prop)->Prop));x04:=(fun (x1:a)=> (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))):(a->Prop)
% Found x00 as proof of ((and ((x R) x04)) (x04 Xx))
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x02))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found conj1:=(conj (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))):(forall (B:Prop), ((forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))->(B->((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) B))))
% Instantiate: b:=(forall (B:Prop), ((forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))->(B->((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) B)))):Prop
% Found conj1 as proof of b
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion0 Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion a) Prop) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found x00:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))
% Instantiate: x:=(fun (x2:(a->(a->Prop))) (x10:(a->Prop))=> (forall (Xx0:a), ((x2 Xx0) Xx0))):((a->(a->Prop))->((a->Prop)->Prop));x02:=(fun (x1:a)=> (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))):(a->Prop)
% Found x00 as proof of ((and ((x R) x02)) (x02 Xx))
% Found x01:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))
% Instantiate: x:=(fun (x2:(a->(a->Prop))) (x10:(a->Prop))=> (forall (Xx0:a), ((x2 Xx0) Xx0))):((a->(a->Prop))->((a->Prop)->Prop));x00:=(fun (x1:a)=> (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))):(a->Prop)
% Found x01 as proof of ((and ((x R) x00)) (x00 Xx))
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found ((eq_ref (a->Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found x00:((and (forall (Xx0:a), ((R Xx0) Xx0))) (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0))))
% Instantiate: x:=(fun (x2:(a->(a->Prop))) (x10:(a->Prop))=> (forall (Xx0:a), ((x2 Xx0) Xx0))):((a->(a->Prop))->((a->Prop)->Prop));x02:=(fun (x1:a)=> (forall (Xx0:a) (Xy:a), (((R Xx0) Xy)->((R Xy) Xx0)))):(a->Prop)
% Found x00 as proof of ((and ((x R) x02)) (x02 Xx))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x00))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> ((((eq_trans000 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((((eq_trans00 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> ((((((eq_trans0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> ((((eq_trans000 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((((eq_trans00 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> ((((((eq_trans0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion0 Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion_dep000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion_dep00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eq_ref00:=(eq_ref0 Xq):(((eq (a->Prop)) Xq) Xq)
% Found (eq_ref0 Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((eq_ref (a->Prop)) Xq) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq)) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->(((eq (a->Prop)) Xq) x00))
% Found (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq)) as proof of (((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->(((eq (a->Prop)) Xq) x00)))
% Found (and_rect00 (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((and_rect0 (((eq (a->Prop)) Xq) x00)) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x0)) (((eq (a->Prop)) Xq) x00)) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((fun (P:Type) (x1:(((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))->((forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))->P)))=> (((((and_rect ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz)))) P) x1) x0)) (((eq (a->Prop)) Xq) x00)) (fun (x1:((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))=> ((eq_ref (a->Prop)) Xq)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref000:=(eq_ref00 P):((P Xq)->(P Xq))
% Found (eq_ref00 P) as proof of ((P Xq)->(P x00))
% Found ((eq_ref0 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eq_ref (a->Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((eq_ref (a->Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x00))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x00))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x00)
% Found x1:(P Xq)
% Instantiate: x00:=Xq:(a->Prop)
% Found (fun (x1:(P Xq))=> x1) as proof of (P x00)
% Found (fun (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of ((P Xq)->(P x00))
% Found (fun (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop)) (x1:(P Xq))=> x1) as proof of (((eq (a->Prop)) Xq) x00)
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> ((((eq_trans000 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((((eq_trans00 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> ((((((eq_trans0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion_dep00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found eta_expansion000:=(eta_expansion00 Xq):(((eq (a->Prop)) Xq) (fun (x:a)=> (Xq x)))
% Found (eta_expansion00 Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found ((eta_expansion0 Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (((eta_expansion a) Prop) Xq) as proof of (forall (P:((a->Prop)->Prop)), ((P Xq)->(P x00)))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((eta_expansion a) Prop) Xq)) as proof of (((eq (a->Prop)) Xq) x00)
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> ((((eq_trans000 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((((eq_trans00 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq 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((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> ((((((eq_trans0 (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (x00:(((eq Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U)))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (R:(a->(a->Prop))), (((and ((and (forall (Xx:a), ((R Xx) Xx))) (forall (Xx:a) (Xy:a), (((R Xx) Xy)->((R Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))->((and (forall (Xp:(a->Prop)), (((x R) Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and ((x R) Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and ((x R) Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))))) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))) (forall (T:(a->(a->Prop))) (U:(a->(a->Prop))), (((and ((and ((and ((and ((and ((and (not (((eq (a->(a->Prop))) T) U))) (forall (Xx:a), ((T Xx) Xx)))) (forall (Xx:a) (Xy:a), (((T Xx) Xy)->((T Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((T Xx) Xy)) ((T Xy) Xz))->((T Xx) Xz))))) (forall (Xx:a), ((U Xx) Xx)))) (forall (Xx:a) (Xy:a), (((U Xx) Xy)->((U Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((U Xx) Xy)) ((U Xy) Xz))->((U Xx) Xz))))->(not (((eq ((a->Prop)->Prop)) (x T)) (x U))))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (P:((a->Prop)->Prop)), (((and (forall (Xp:(a->Prop)), ((P Xp)->((ex a) (fun (Xz:a)=> (Xp Xz)))))) (forall (Xx:a), ((ex (a->Prop)) (fun (Xp:(a->Prop))=> ((and ((and (P Xp)) (Xp Xx))) (forall (Xq:(a->Prop)), (((and (P Xq)) (Xq Xx))->(((eq (a->Prop)) Xq) Xp))))))))->((ex (a->(a->Prop))) (fun (S:(a->(a->Prop)))=> ((and ((and ((and (forall (Xx:a), ((S Xx) Xx))) (forall (Xx:a) (Xy:a), (((S Xx) Xy)->((S Xy) Xx))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((S Xx) Xy)) ((S Xy) Xz))->((S Xx) Xz))))) (((eq ((a->Prop)->Prop)) P) (x S))))))))
% Found eta_expansion0000:=(eta_expansion000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion0 Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion a) Prop) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion a) Prop) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion0 Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion a) Prop) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x02)
% Found x10:(P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of (P Xq)
% Found (fun (x10:(P Xq))=> x10) as proof of ((P Xq)->(P Xq))
% Found (eta_expansion_dep0000 (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion_dep000 (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion_dep00 Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10)) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> (((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) (fun (x2:(a->Prop))=> ((P Xq)->(P x2)))) (fun (x10:(P Xq))=> x10))) as proof of (((eq (a->Prop)) Xq) x02)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P Xq)->(P (fun (x:a)=> (Xq x))))
% Found (eta_expansion_dep000 P) as proof of ((P Xq)->(P x02))
% Found ((eta_expansion_dep00 Xq) P) as proof of ((P Xq)->(P x02))
% Found (((eta_expansion_dep0 (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P) as proof of ((P Xq)->(P x02))
% Found (fun (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of ((P Xq)->(P x02))
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx))) (P:((a->Prop)->Prop))=> ((((eta_expansion_dep a) (fun (x2:a)=> Prop)) Xq) P)) as proof of (((eq (a->Prop)) Xq) x02)
% Found eq_ref00:=(eq_ref0 x02):(((eq (a->Prop)) x02) x02)
% Found (eq_ref0 x02) as proof of (((eq (a->Prop)) x02) Xq)
% Found ((eq_ref (a->Prop)) x02) as proof of (((eq (a->Prop)) x02) Xq)
% Found ((eq_ref (a->Prop)) x02) as proof of (((eq (a->Prop)) x02) Xq)
% Found ((eq_ref (a->Prop)) x02) as proof of (((eq (a->Prop)) x02) Xq)
% Found (eq_sym000 ((eq_ref (a->Prop)) x02)) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((eq_sym00 Xq) ((eq_ref (a->Prop)) x02)) as proof of (((eq (a->Prop)) Xq) x02)
% Found (((eq_sym0 x02) Xq) ((eq_ref (a->Prop)) x02)) as proof of (((eq (a->Prop)) Xq) x02)
% Found ((((eq_sym (a->Prop)) x02) Xq) ((eq_ref (a->Prop)) x02)) as proof of (((eq (a->Prop)) Xq) x02)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> ((((eq_sym (a->Prop)) x02) Xq) ((eq_ref (a->Prop)) x02))) as proof of (((eq (a->Prop)) Xq) x02)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x00 x)))
% Found (functional_extensionality0000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((functional_extensionality000 x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((functional_extensionality00 Xq) x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((((functional_extensionality0 Prop) Xq) x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((((functional_extensionality a) Prop) Xq) x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (fun (x000:((and ((x R) Xq)) (Xq Xx)))=> (((((functional_extensionality a) Prop) Xq) x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1))))) as proof of (((eq (a->Prop)) Xq) x00)
% Found eq_ref00:=(eq_ref0 (Xq x1)):(((eq Prop) (Xq x1)) (Xq x1))
% Found (eq_ref0 (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found ((eq_ref Prop) (Xq x1)) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (((eq Prop) (Xq x1)) (x00 x1))
% Found (fun (x1:a)=> ((eq_ref Prop) (Xq x1))) as proof of (forall (x:a), (((eq Prop) (Xq x)) (x00 x)))
% Found (functional_extensionality_dep0000 (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found ((functional_extensionality_dep000 x00) (fun (x1:a)=> ((eq_ref Prop) (Xq x1)))) as proof of (((eq (a->Prop)) Xq) x00)
% Found (((functional_extensionality_dep00 Xq) x00) (fun (x1:a)=> ((eq_re
% EOF
%------------------------------------------------------------------------------