TSTP Solution File: PUZ093^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : PUZ093^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n100.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 13:28:58 EDT 2014
% Result : Timeout 300.04s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : PUZ093^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n100.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:15:46 CDT 2014
% % CPUTime : 300.04
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula ((ex fofType) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) of role conjecture named cSIXFRIENDS_PTH
% Conjecture to prove = ((ex fofType) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['((ex fofType) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))))']
% Parameter fofType:Type.
% Trying to prove ((ex fofType) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))):(((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))))
% Found (eta_expansion00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xa)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P Xa)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P Xa)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P Xa)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))):(((eq (fofType->Prop)) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))) (fun (x0:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P x0))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P x0))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P x0))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P x0)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))
% Found (eta_expansion00 (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))) as proof of (((eq (fofType->Prop)) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))) as proof of (((eq (fofType->Prop)) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))) as proof of (((eq (fofType->Prop)) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))) as proof of (((eq (fofType->Prop)) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))) as proof of (((eq (fofType->Prop)) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))) 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) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found (((eq_trans000 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((((eq_trans00 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found (((((eq_trans0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((((((eq_trans Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% 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) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found (((eq_trans000 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((((eq_trans00 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found (((((eq_trans0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((((((eq_trans Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% 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 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))):(((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found (eq_ref0 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) 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 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))):(((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found (eq_ref0 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) b)
% Found ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) 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 (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) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x))
% Found ((((eq_trans00 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x))
% Found (((((eq_trans0 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x))
% Found ((((((eq_trans Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x))
% Found ((((((eq_trans Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x))
% Found (eq_sym000 ((((((eq_trans Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_sym00 (f x)) ((((((eq_trans Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found (((eq_sym0 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((((((eq_trans Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((((eq_sym Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((((((eq_trans Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) ((((eq_sym Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% 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) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x))
% Found ((((eq_trans00 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x))
% Found (((((eq_trans0 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x))
% Found ((((((eq_trans Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x))
% Found ((((((eq_trans Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x))
% Found (eq_sym000 ((((((eq_trans Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_sym00 (f x)) ((((((eq_trans Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found (((eq_sym0 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((((((eq_trans Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) (((eq_sym0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((((eq_sym Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((((((eq_trans Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) (f x)) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) ((((eq_sym Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% 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) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))))
% Found (((eq_trans000 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))))
% Found ((((eq_trans00 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))))
% Found (((((eq_trans0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))))
% Found ((((((eq_trans Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))))
% Found ((((((eq_trans Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))))
% 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) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))))
% Found (((eq_trans000 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))))
% Found ((((eq_trans00 ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))))
% Found (((((eq_trans0 (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))))
% Found ((((((eq_trans Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))))
% Found ((((((eq_trans Prop) (f x)) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh))))))))))))))))))))))))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((ex fofType) (fun (Xaa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xbb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((ex fofType) (fun (Xcc:fofType)=> ((ex fofType) (fun (Xd:fofType)=> ((ex fofType) (fun (Xdd:fofType)=> ((ex fofType) (fun (Xe:fofType)=> ((ex fofType) (fun (Xee:fofType)=> ((ex fofType) (fun (Xh:fofType)=> ((ex fofType) (fun (Xhh:fofType)=> (forall (P:(fofType->fofType)), (((and ((and ((and ((and ((and (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xb)) (P Xbb)))) (((eq fofType) (P Xe)) (P Xhh)))->(((eq fofType) (P Xc)) (P Xdd)))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xb)) (P Xcc)))->(not (((eq fofType) (P Xd)) (P Xee)))))) (((and ((and ((and (((eq fofType) (P Xc)) (P Xcc))) (((eq fofType) (P Xcc)) (P Xd)))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P x)) (P Xbb))))->(not (((eq fofType) (P Xe)) (P Xhh)))))) (((and ((and (((eq fofType) (P x)) (P Xaa))) (((eq fofType) (P Xd)) (P Xdd)))) (not (((eq fofType) (P Xb)) (P Xcc))))->(((eq fofType) (P Xe)) (P Xhh))))) (((and ((and (((eq fofType) (P Xe)) (P Xee))) (((eq fofType) (P Xh)) (P Xhh)))) (((eq fofType) (P Xc)) (P Xdd)))->(not (((eq fofType) (P x)) (P Xbb)))))) (((and ((and ((and (((eq fofType) (P Xb)) (P Xbb))) (((eq fofType) (P Xbb)) (P Xc)))) (((eq fofType) (P Xc)) (P Xcc)))) (not (((eq fofType) (P Xe)) (P Xhh))))->(((eq fofType) (P Xd)) (P Xee))))->((or ((or ((or ((or ((or (not (((eq fofType) (P x)) (P Xaa)))) (not (((eq fofType) (P Xb)) (P Xbb))))) (not (((eq fofType) (P Xc)) (P Xcc))))) (not (((eq fofType) (P Xd)) (P Xdd))))) (not (((eq fofType) (P Xe)) (P Xee))))) (not (((eq fofType) (P Xh)) (P Xhh)))))))))))))))))))))))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((ex fofType) (fu
% EOF
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