%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV250^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n091.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:57 EDT 2014

% Result   : Timeout 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV250^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n091.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:37:31 CDT 2014
% % CPUTime  : 300.02 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x1992680>, <kernel.DependentProduct object at 0x1992cf8>) of role type named cOPEN
% Using role type
% Declaring cOPEN:((fofType->Prop)->Prop)
% FOF formula ((forall (D:(fofType->Prop)) (G:((fofType->Prop)->Prop)), (((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) D) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))->(cOPEN D)))->(forall (A:(fofType->Prop)), ((ex (fofType->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))))) of role conjecture named cEXISTS_INTERIOR_pme
% Conjecture to prove = ((forall (D:(fofType->Prop)) (G:((fofType->Prop)->Prop)), (((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) D) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))->(cOPEN D)))->(forall (A:(fofType->Prop)), ((ex (fofType->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['((forall (D:(fofType->Prop)) (G:((fofType->Prop)->Prop)), (((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) D) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))->(cOPEN D)))->(forall (A:(fofType->Prop)), ((ex (fofType->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx))))))))))']
% Parameter fofType:Type.
% Parameter cOPEN:((fofType->Prop)->Prop).
% Trying to prove ((forall (D:(fofType->Prop)) (G:((fofType->Prop)->Prop)), (((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) D) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))->(cOPEN D)))->(forall (A:(fofType->Prop)), ((ex (fofType->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx))))))))))
% Found eq_ref000:=(eq_ref00 C):((C Xx)->(C Xx))
% Found (eq_ref00 C) as proof of ((C Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found x000:(C Xx)
% Instantiate: x0:=C:(fofType->Prop)
% Found (fun (x000:(C Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:fofType) (x000:(C Xx))=> x000) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType) (x000:(C Xx))=> x000) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eta_expansion000:=(eta_expansion00 (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))):(((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) (fun (x:(fofType->Prop))=> ((and ((and (cOPEN x)) (forall (Xx:fofType), ((x Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x Xx))))))))
% Found (eta_expansion00 (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) b)
% Found ((eta_expansion0 Prop) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) b)
% Found (((eta_expansion (fofType->Prop)) Prop) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) Xx)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) Xx)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) Xx)
% Found (eq_sym0000 ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found (eq_sym0000 ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x1:(((eq fofType) Xx) Xx))=> ((eq_sym000 x1) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x1:(((eq fofType) Xx) Xx))=> (((eq_sym00 Xx) x1) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x1:(((eq fofType) Xx) Xx))=> ((((eq_sym0 Xx) Xx) x1) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x1:(((eq fofType) Xx) Xx))=> (((((eq_sym fofType) Xx) Xx) x1) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x1:(((eq fofType) Xx) Xx))=> (((((eq_sym fofType) Xx) Xx) x1) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> ((fun (x1:(((eq fofType) Xx) Xx))=> (((((eq_sym fofType) Xx) Xx) x1) C)) ((eq_ref fofType) Xx))) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> ((fun (x1:(((eq fofType) Xx) Xx))=> (((((eq_sym fofType) Xx) Xx) x1) C)) ((eq_ref fofType) Xx))) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found ((conj20 (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (((conj2 (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0))) as proof of (cOPEN x0)
% Found ((x1 cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (cOPEN x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (cOPEN x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (cOPEN x0)
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found eq_ref00:=(eq_ref0 b):(((eq ((fofType->Prop)->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx))))))))
% Found ((eq_ref ((fofType->Prop)->Prop)) b) as proof of (((eq ((fofType->Prop)->Prop)) b) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx))))))))
% Found eq_ref00:=(eq_ref0 a):(((eq ((fofType->Prop)->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) a) as proof of (((eq ((fofType->Prop)->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))):(((eq Prop) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found (eq_ref0 (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))) b)
% Found ((eq_ref Prop) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))) as proof of (((eq Prop) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))) b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found x000:(C Xx)
% Instantiate: x0:=C:(fofType->Prop)
% Found (fun (x000:(C Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:fofType) (x000:(C Xx))=> x000) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType) (x000:(C Xx))=> x000) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 C):((C Xx)->(C Xx))
% Found (eq_ref00 C) as proof of ((C Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 C):((C Xx)->(C Xx))
% Found (eq_ref00 C) as proof of ((C Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x1:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found x2:(C Xx)
% Instantiate: x0:=C:(fofType->Prop)
% Found (fun (x2:(C Xx))=> x2) as proof of (x0 Xx)
% Found (fun (Xx:fofType) (x2:(C Xx))=> x2) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x1:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType) (x2:(C Xx))=> x2) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((eq_trans000 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((((eq_trans0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((eq_trans000 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((((eq_trans0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eq_ref00:=(eq_ref0 (forall (Xx:fofType), ((x0 Xx)->(A Xx)))):(((eq Prop) (forall (Xx:fofType), ((x0 Xx)->(A Xx)))) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))
% Found (eq_ref0 (forall (Xx:fofType), ((x0 Xx)->(A Xx)))) as proof of (((eq Prop) (forall (Xx:fofType), ((x0 Xx)->(A Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType), ((x0 Xx)->(A Xx)))) as proof of (((eq Prop) (forall (Xx:fofType), ((x0 Xx)->(A Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType), ((x0 Xx)->(A Xx)))) as proof of (((eq Prop) (forall (Xx:fofType), ((x0 Xx)->(A Xx)))) b)
% Found ((eq_ref Prop) (forall (Xx:fofType), ((x0 Xx)->(A Xx)))) as proof of (((eq Prop) (forall (Xx:fofType), ((x0 Xx)->(A Xx)))) b)
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: b:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of b
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) b)
% Found eq_ref00:=(eq_ref0 b):(((eq fofType) b) b)
% Found (eq_ref0 b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_ref fofType) b) as proof of (((eq fofType) b) Xx)
% Found ((eq_trans00000 ((eq_ref fofType) Xx)) ((eq_ref fofType) b)) as proof of ((C Xx)->(x0 Xx))
% Found ((eq_trans00000 ((eq_ref fofType) Xx)) ((eq_ref fofType) b)) as proof of ((C Xx)->(x0 Xx))
% Found (((fun (x1:(((eq fofType) Xx) b)) (x2:(((eq fofType) b) Xx))=> (((eq_trans0000 x1) x2) C)) ((eq_ref fofType) Xx)) ((eq_ref fofType) b)) as proof of ((C Xx)->(x0 Xx))
% Found (((fun (x1:(((eq fofType) Xx) Xx)) (x2:(((eq fofType) Xx) Xx))=> ((((eq_trans000 Xx) x1) x2) C)) ((eq_ref fofType) Xx)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found (((fun (x1:(((eq fofType) Xx) Xx)) (x2:(((eq fofType) Xx) Xx))=> (((((fun (b:fofType)=> ((eq_trans00 b) Xx)) Xx) x1) x2) C)) ((eq_ref fofType) Xx)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found (((fun (x1:(((eq fofType) Xx) Xx)) (x2:(((eq fofType) Xx) Xx))=> (((((fun (b:fofType)=> (((eq_trans0 Xx) b) Xx)) Xx) x1) x2) C)) ((eq_ref fofType) Xx)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found (((fun (x1:(((eq fofType) Xx) Xx)) (x2:(((eq fofType) Xx) Xx))=> (((((fun (b:fofType)=> ((((eq_trans fofType) Xx) b) Xx)) Xx) x1) x2) C)) ((eq_ref fofType) Xx)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found (((fun (x1:(((eq fofType) Xx) Xx)) (x2:(((eq fofType) Xx) Xx))=> (((((fun (b:fofType)=> ((((eq_trans fofType) Xx) b) Xx)) Xx) x1) x2) C)) ((eq_ref fofType) Xx)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> (((fun (x1:(((eq fofType) Xx) Xx)) (x2:(((eq fofType) Xx) Xx))=> (((((fun (b:fofType)=> ((((eq_trans fofType) Xx) b) Xx)) Xx) x1) x2) C)) ((eq_ref fofType) Xx)) ((eq_ref fofType) Xx))) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> (((fun (x1:(((eq fofType) Xx) Xx)) (x2:(((eq fofType) Xx) Xx))=> (((((fun (b:fofType)=> ((((eq_trans fofType) Xx) b) Xx)) Xx) x1) x2) C)) ((eq_ref fofType) Xx)) ((eq_ref fofType) Xx))) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of b
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (x0 x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (x0 x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) (x0 x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (x0 x1))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) (x0 x1))
% Found (fun (x1:fofType)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:fofType), (((eq Prop) (f x)) (x0 x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found (((eq_trans000 (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found (((((eq_trans0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found (eq_sym000 ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_sym00 (f x0)) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((eq_sym0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((((eq_sym Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) ((((eq_sym Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found x2:(C Xx)
% Instantiate: x0:=C:(fofType->Prop)
% Found (fun (x2:(C Xx))=> x2) as proof of (x0 Xx)
% Found (fun (Xx:fofType) (x2:(C Xx))=> x2) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x1:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType) (x2:(C Xx))=> x2) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 C):((C Xx)->(C Xx))
% Found (eq_ref00 C) as proof of ((C Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x1:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found (((eq_trans000 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found ((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found (((((eq_trans0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found (((eq_trans000 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found ((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found (((((eq_trans0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref000:=(eq_ref00 C):((C Xx)->(C Xx))
% Found (eq_ref00 C) as proof of ((C Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found x000:(C Xx)
% Instantiate: x0:=C:(fofType->Prop)
% Found (fun (x000:(C Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:fofType) (x000:(C Xx))=> x000) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType) (x000:(C Xx))=> x000) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 C):((C Xx)->(C Xx))
% Found (eq_ref00 C) as proof of ((C Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found x000:(C Xx)
% Instantiate: x0:=C:(fofType->Prop)
% Found (fun (x000:(C Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:fofType) (x000:(C Xx))=> x000) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType) (x000:(C Xx))=> x000) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (b x)))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: a:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of a
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (b x)))
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: b:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of b
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eta_expansion000:=(eta_expansion00 b):(((eq ((fofType->Prop)->Prop)) b) (fun (x:(fofType->Prop))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% Found (((eta_expansion (fofType->Prop)) Prop) b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% Found (((eta_expansion (fofType->Prop)) Prop) b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% Found (((eta_expansion (fofType->Prop)) Prop) b) as proof of (((eq ((fofType->Prop)->Prop)) b) b0)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq Prop) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq Prop) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq Prop) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq Prop) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found ((conj00 (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (((conj0 (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0))) as proof of (b x0)
% Found ((x1 cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (b x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (b x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (b x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (b x0)
% Found (ex_intro000 (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0)))) as proof of (P b)
% Found ((ex_intro00 (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (((x (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5))))))))) as proof of (P b)
% Found (((ex_intro0 b) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (((x (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5))))))))) as proof of (P b)
% Found ((((ex_intro (fofType->Prop)) b) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (((x (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5))))))))) as proof of (P b)
% Found ((((ex_intro (fofType->Prop)) b) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (((x (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5))))))))) as proof of (P b)
% Found eq_ref000:=(eq_ref00 P):((P (x0 x2))->(P (x0 x2)))
% Found (eq_ref00 P) as proof of (P0 (x0 x2))
% Found ((eq_ref0 (x0 x2)) P) as proof of (P0 (x0 x2))
% Found (((eq_ref Prop) (x0 x2)) P) as proof of (P0 (x0 x2))
% Found (((eq_ref Prop) (x0 x2)) P) as proof of (P0 (x0 x2))
% Found eq_ref000:=(eq_ref00 P):((P (x0 x2))->(P (x0 x2)))
% Found (eq_ref00 P) as proof of (P0 (x0 x2))
% Found ((eq_ref0 (x0 x2)) P) as proof of (P0 (x0 x2))
% Found (((eq_ref Prop) (x0 x2)) P) as proof of (P0 (x0 x2))
% Found (((eq_ref Prop) (x0 x2)) P) as proof of (P0 (x0 x2))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of b
% Found eq_ref000:=(eq_ref00 P):((P (x0 x2))->(P (x0 x2)))
% Found (eq_ref00 P) as proof of (P0 (x0 x2))
% Found ((eq_ref0 (x0 x2)) P) as proof of (P0 (x0 x2))
% Found (((eq_ref Prop) (x0 x2)) P) as proof of (P0 (x0 x2))
% Found (((eq_ref Prop) (x0 x2)) P) as proof of (P0 (x0 x2))
% Found eq_ref000:=(eq_ref00 P):((P (x0 x2))->(P (x0 x2)))
% Found (eq_ref00 P) as proof of (P0 (x0 x2))
% Found ((eq_ref0 (x0 x2)) P) as proof of (P0 (x0 x2))
% Found (((eq_ref Prop) (x0 x2)) P) as proof of (P0 (x0 x2))
% Found (((eq_ref Prop) (x0 x2)) P) as proof of (P0 (x0 x2))
% Found eq_ref000:=(eq_ref00 C):((C Xx)->(C Xx))
% Found (eq_ref00 C) as proof of ((C Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found x000:(C Xx)
% Instantiate: x0:=C:(fofType->Prop)
% Found (fun (x000:(C Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:fofType) (x000:(C Xx))=> x000) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType) (x000:(C Xx))=> x000) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eq_ref000:=(eq_ref00 C):((C Xx)->(C Xx))
% Found (eq_ref00 C) as proof of ((C Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x1:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found x2:(C Xx)
% Instantiate: x0:=C:(fofType->Prop)
% Found (fun (x2:(C Xx))=> x2) as proof of (x0 Xx)
% Found (fun (Xx:fofType) (x2:(C Xx))=> x2) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x1:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType) (x2:(C Xx))=> x2) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 f):(((eq ((fofType->Prop)->Prop)) f) (fun (x:(fofType->Prop))=> (f x)))
% Found (eta_expansion_dep00 f) as proof of (((eq ((fofType->Prop)->Prop)) f) b)
% Found ((eta_expansion_dep0 (fun (x1:(fofType->Prop))=> Prop)) f) as proof of (((eq ((fofType->Prop)->Prop)) f) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) f) as proof of (((eq ((fofType->Prop)->Prop)) f) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) f) as proof of (((eq ((fofType->Prop)->Prop)) f) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x1:(fofType->Prop))=> Prop)) f) as proof of (((eq ((fofType->Prop)->Prop)) f) b)
% Found eq_ref00:=(eq_ref0 f):(((eq ((fofType->Prop)->Prop)) f) f)
% Found (eq_ref0 f) as proof of (((eq ((fofType->Prop)->Prop)) f) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) f) as proof of (((eq ((fofType->Prop)->Prop)) f) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) f) as proof of (((eq ((fofType->Prop)->Prop)) f) b)
% Found ((eq_ref ((fofType->Prop)->Prop)) f) as proof of (((eq ((fofType->Prop)->Prop)) f) b)
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found x1:(P0 (f x0))
% Instantiate: b:=(f x0):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found x1:(P0 (f x0))
% Instantiate: b:=(f x0):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((eq_trans0000 x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_trans000 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((((eq_trans0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((eq_trans0000 x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_trans000 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((((eq_trans0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((eq_trans0000 x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_trans000 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((((eq_trans0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((eq_trans0000 x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_trans000 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((((eq_trans0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x0)):(((eq Prop) (f0 x0)) (f0 x0))
% Found (eq_ref0 (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found ((eq_ref Prop) (f0 x0)) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f0 x0))) as proof of (((eq Prop) (f0 x0)) (f x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f0 x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found ((conj20 (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (((conj2 (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0))) as proof of (cOPEN x0)
% Found ((x1 cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0))) as proof of (cOPEN x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0))) as proof of (cOPEN x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0))) as proof of (cOPEN x0)
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found ((conj20 (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (((conj2 (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) x0))) as proof of (cOPEN x0)
% Found ((x1 cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) ((eq_ref (fofType->Prop)) x0))) as proof of (cOPEN x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) ((eq_ref (fofType->Prop)) x0))) as proof of (cOPEN x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) ((eq_ref (fofType->Prop)) x0))) as proof of (cOPEN x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))):(((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) (fun (x:(fofType->Prop))=> ((and ((and (cOPEN x)) (forall (Xx:fofType), ((x Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x Xx))))))))
% Found (eta_expansion_dep00 (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) b)
% Found ((eta_expansion_dep0 (fun (x3:(fofType->Prop))=> Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) b)
% Found (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) as proof of (((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx)))))))) b)
% Found (eq_sym0001 (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx))))))))) as proof of ((C Xx)->(x0 Xx))
% Found (eq_sym0001 (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx:fofType), ((B Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(B Xx))))))))) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x2:(((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx0:fofType), ((B Xx0)->(A Xx0))))) (forall (C0:(fofType->Prop)), (((and (cOPEN C0)) (forall (Xx0:fofType), ((C0 Xx0)->(A Xx0))))->(forall (Xx0:fofType), ((C0 Xx0)->(B Xx0)))))))) b))=> ((eq_sym000 x2) (fun (x3:((fofType->Prop)->Prop))=> (C Xx)))) (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx0:fofType), ((B Xx0)->(A Xx0))))) (forall (C0:(fofType->Prop)), (((and (cOPEN C0)) (forall (Xx0:fofType), ((C0 Xx0)->(A Xx0))))->(forall (Xx0:fofType), ((C0 Xx0)->(B Xx0))))))))) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x2:(((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx0:fofType), ((B Xx0)->(A Xx0))))) (forall (C0:(fofType->Prop)), (((and (cOPEN C0)) (forall (Xx0:fofType), ((C0 Xx0)->(A Xx0))))->(forall (Xx0:fofType), ((C0 Xx0)->(B Xx0)))))))) b))=> ((eq_sym000 x2) (fun (x3:((fofType->Prop)->Prop))=> (C Xx)))) (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx0:fofType), ((B Xx0)->(A Xx0))))) (forall (C0:(fofType->Prop)), (((and (cOPEN C0)) (forall (Xx0:fofType), ((C0 Xx0)->(A Xx0))))->(forall (Xx0:fofType), ((C0 Xx0)->(B Xx0))))))))) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> ((fun (x2:(((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx0:fofType), ((B Xx0)->(A Xx0))))) (forall (C0:(fofType->Prop)), (((and (cOPEN C0)) (forall (Xx0:fofType), ((C0 Xx0)->(A Xx0))))->(forall (Xx0:fofType), ((C0 Xx0)->(B Xx0)))))))) b))=> ((eq_sym000 x2) (fun (x3:((fofType->Prop)->Prop))=> (C Xx)))) (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx0:fofType), ((B Xx0)->(A Xx0))))) (forall (C0:(fofType->Prop)), (((and (cOPEN C0)) (forall (Xx0:fofType), ((C0 Xx0)->(A Xx0))))->(forall (Xx0:fofType), ((C0 Xx0)->(B Xx0)))))))))) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x1:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> ((fun (x2:(((eq ((fofType->Prop)->Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx0:fofType), ((B Xx0)->(A Xx0))))) (forall (C0:(fofType->Prop)), (((and (cOPEN C0)) (forall (Xx0:fofType), ((C0 Xx0)->(A Xx0))))->(forall (Xx0:fofType), ((C0 Xx0)->(B Xx0)))))))) b))=> ((eq_sym000 x2) (fun (x3:((fofType->Prop)->Prop))=> (C Xx)))) (((eta_expansion_dep (fofType->Prop)) (fun (x3:(fofType->Prop))=> Prop)) (fun (B:(fofType->Prop))=> ((and ((and (cOPEN B)) (forall (Xx0:fofType), ((B Xx0)->(A Xx0))))) (forall (C0:(fofType->Prop)), (((and (cOPEN C0)) (forall (Xx0:fofType), ((C0 Xx0)->(A Xx0))))->(forall (Xx0:fofType), ((C0 Xx0)->(B Xx0)))))))))) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found ex_intro0:=(ex_intro (fofType->Prop)):(forall (P:((fofType->Prop)->Prop)) (x:(fofType->Prop)), ((P x)->((ex (fofType->Prop)) P)))
% Instantiate: a:=(forall (P:((fofType->Prop)->Prop)) (x:(fofType->Prop)), ((P x)->((ex (fofType->Prop)) P))):Prop
% Found ex_intro0 as proof of a
% Found x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Instantiate: b:=((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_sym0100 ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found ((eq_sym0100 ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found (((fun (x2:(((eq Prop) (f x0)) b))=> ((eq_sym010 x2) P0)) ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((eq_sym01 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found (fun (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1)) as proof of (P0 (f x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1)) as proof of ((P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))->(P0 (f x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1)) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found (eq_sym000 (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_sym00 (f x0)) (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((eq_sym0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((((eq_sym Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((eq_sym Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (x0 x2))
% Found eq_ref00:=(eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))):(((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found ((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) as proof of (((eq Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (fofType->Prop)) b) (fun (x:fofType)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref000:=(eq_ref00 P):((P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))->(P ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))))
% Found (eq_ref00 P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found (((eq_ref Prop) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2))))) P) as proof of (P0 ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) Xx)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) Xx)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) Xx)
% Found (eq_sym1000 ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found (eq_sym1000 ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x1:(((eq fofType) Xx) Xx))=> ((eq_sym100 x1) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x1:(((eq fofType) Xx) Xx))=> (((eq_sym10 Xx) x1) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x1:(((eq fofType) Xx) Xx))=> ((((eq_sym1 Xx) Xx) x1) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x1:(((eq fofType) Xx) Xx))=> (((((eq_sym fofType) Xx) Xx) x1) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x1:(((eq fofType) Xx) Xx))=> (((((eq_sym fofType) Xx) Xx) x1) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> ((fun (x1:(((eq fofType) Xx) Xx))=> (((((eq_sym fofType) Xx) Xx) x1) C)) ((eq_ref fofType) Xx))) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> ((fun (x1:(((eq fofType) Xx) Xx))=> (((((eq_sym fofType) Xx) Xx) x1) C)) ((eq_ref fofType) Xx))) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eq_ref00:=(eq_ref0 Xx):(((eq fofType) Xx) Xx)
% Found (eq_ref0 Xx) as proof of (((eq fofType) Xx) Xx)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) Xx)
% Found ((eq_ref fofType) Xx) as proof of (((eq fofType) Xx) Xx)
% Found (eq_sym1000 ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found (eq_sym1000 ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x2:(((eq fofType) Xx) Xx))=> ((eq_sym100 x2) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x2:(((eq fofType) Xx) Xx))=> (((eq_sym10 Xx) x2) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x2:(((eq fofType) Xx) Xx))=> ((((eq_sym1 Xx) Xx) x2) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x2:(((eq fofType) Xx) Xx))=> (((((eq_sym fofType) Xx) Xx) x2) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found ((fun (x2:(((eq fofType) Xx) Xx))=> (((((eq_sym fofType) Xx) Xx) x2) C)) ((eq_ref fofType) Xx)) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> ((fun (x2:(((eq fofType) Xx) Xx))=> (((((eq_sym fofType) Xx) Xx) x2) C)) ((eq_ref fofType) Xx))) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x1:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> ((fun (x2:(((eq fofType) Xx) Xx))=> (((((eq_sym fofType) Xx) Xx) x2) C)) ((eq_ref fofType) Xx))) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found ((conj00 (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (((conj0 (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0))) as proof of (f x0)
% Found ((x1 cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (f x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (f x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (f x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (f x0)
% Found (ex_intro000 (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0)))) as proof of (P f)
% Found ((ex_intro00 (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (((x (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5))))))))) as proof of (P f)
% Found (((ex_intro0 f) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (((x (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5))))))))) as proof of (P f)
% Found ((((ex_intro (fofType->Prop)) f) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (((x (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5))))))))) as proof of (P f)
% Found ((((ex_intro (fofType->Prop)) f) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (((x (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5))))))))) as proof of (P f)
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found x1:(P0 (f x0))
% Instantiate: b:=(f x0):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Instantiate: b:=((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref Prop) b) P0) as proof of (P1 b)
% Found (((eq_ref Prop) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) P0) as proof of (P1 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) P0) as proof of (P1 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) P0) as proof of (P1 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found (((eq_trans000 (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found (((((eq_trans0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((eq_sym0000 ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((eq_sym0000 ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> ((eq_sym000 x1) (fun (x3:Prop)=> ((P0 (f x0))->(P0 x3))))) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> (((eq_sym00 (f x0)) x1) (fun (x3:Prop)=> ((P0 (f x0))->(P0 x3))))) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> ((((eq_sym0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) x1) (fun (x3:Prop)=> ((P0 (f x0))->(P0 x3))))) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> (((((eq_sym Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) x1) (fun (x3:Prop)=> ((P0 (f x0))->(P0 x3))))) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) ((((eq_sym Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> (((((eq_sym Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) x1) (fun (x3:Prop)=> ((P0 (f x0))->(P0 x3))))) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) ((((eq_sym Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref Prop) b) P0) as proof of (P1 b)
% Found (((eq_ref Prop) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eta_expansion000:=(eta_expansion00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion00 x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion0 Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion fofType) Prop) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found ((conj00 (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (((conj0 (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion fofType) Prop) x0))) as proof of (f x0)
% Found ((x1 cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (f x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (f x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (f x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0))) as proof of (f x0)
% Found (ex_intro000 (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) x0)))) as proof of (P f)
% Found ((ex_intro00 (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (((x (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5))))))))) as proof of (P f)
% Found (((ex_intro0 f) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (((x (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5))))))))) as proof of (P f)
% Found ((((ex_intro (fofType->Prop)) f) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (((x (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5))))))))) as proof of (P f)
% Found ((((ex_intro (fofType->Prop)) f) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (((x (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5)))))) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion fofType) Prop) (fun (x5:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S x5))))))))) as proof of (P f)
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found x1:(P0 (f x0))
% Instantiate: b:=(f x0):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Instantiate: b:=((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref Prop) b) P0) as proof of (P1 b)
% Found (((eq_ref Prop) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (eq_ref00 P0) as proof of (P1 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) P0) as proof of (P1 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) P0) as proof of (P1 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) P0) as proof of (P1 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) x0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found (((eq_trans000 (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found (((((eq_trans0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found (eq_sym0000 ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (eq_sym0000 ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> ((eq_sym000 x1) P0)) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> (((eq_sym00 (f x0)) x1) P0)) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> ((((eq_sym0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) x1) P0)) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> (((((eq_sym Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) x1) P0)) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) ((((eq_sym Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> (((((eq_sym Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) x1) P0)) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) ((((eq_sym Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0))) as proof of (((eq Prop) b) (f x0))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found (((eq_trans000 (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) b) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found (((((eq_trans0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0)))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found ((eq_sym0000 ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((eq_sym0000 ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> ((eq_sym000 x1) (fun (x3:Prop)=> ((P0 (f x0))->(P0 x3))))) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> (((eq_sym00 (f x0)) x1) (fun (x3:Prop)=> ((P0 (f x0))->(P0 x3))))) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> ((((eq_sym0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) x1) (fun (x3:Prop)=> ((P0 (f x0))->(P0 x3))))) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> (((((eq_sym Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) x1) (fun (x3:Prop)=> ((P0 (f x0))->(P0 x3))))) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) ((((eq_sym Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x1:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)))=> (((((eq_sym Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) x1) (fun (x3:Prop)=> ((P0 (f x0))->(P0 x3))))) ((((((eq_trans Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) ((((eq_sym Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((eq_ref Prop) (f x0))))) (((eq_ref Prop) (f x0)) P0))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 b)->(P0 b))
% Found (eq_ref00 P0) as proof of (P1 b)
% Found ((eq_ref0 b) P0) as proof of (P1 b)
% Found (((eq_ref Prop) b) P0) as proof of (P1 b)
% Found (((eq_ref Prop) b) P0) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 P):((P x0)->(P x0))
% Found (eq_ref00 P) as proof of (P0 x0)
% Found ((eq_ref0 x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found (((eq_ref (fofType->Prop)) x0) P) as proof of (P0 x0)
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of a
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion 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 eq_ref00:=(eq_ref0 (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))):(((eq Prop) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (eq_ref0 (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) as proof of (((eq Prop) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) b)
% Found ((eq_ref Prop) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) as proof of (((eq Prop) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) b)
% Found ((eq_ref Prop) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) as proof of (((eq Prop) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) b)
% Found ((eq_ref Prop) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) as proof of (((eq Prop) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) b)
% Found x1:(P0 (f x0))
% Instantiate: b:=(f x0):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found x1:(P0 (f x0))
% Instantiate: b:=(f x0):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found x1:(P0 (f x0))
% Instantiate: b:=(f x0):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found x1:(P0 (f x0))
% Instantiate: b:=(f x0):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 x0):(((eq (fofType->Prop)) x0) (fun (x:fofType)=> (x0 x)))
% Found (eta_expansion_dep00 x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found ((conj20 (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (((conj2 (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0))) as proof of (cOPEN x0)
% Found ((x1 cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0))) as proof of (cOPEN x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0))) as proof of (cOPEN x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) x0))) as proof of (cOPEN x0)
% Found eq_ref00:=(eq_ref0 x0):(((eq (fofType->Prop)) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref (fofType->Prop)) x0) as proof of (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found ((conj20 (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (((conj2 (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) x0)) as proof of ((and (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (x10 ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) ((eq_ref (fofType->Prop)) x0))) as proof of (cOPEN x0)
% Found ((x1 cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) ((eq_ref (fofType->Prop)) x0))) as proof of (cOPEN x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) ((eq_ref (fofType->Prop)) x0))) as proof of (cOPEN x0)
% Found (((x x0) cOPEN) ((((conj (forall (Xx:(fofType->Prop)), ((cOPEN Xx)->(cOPEN Xx)))) (((eq (fofType->Prop)) x0) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (cOPEN S)) (S Xx))))))) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) cOPEN))) ((eq_ref (fofType->Prop)) x0))) as proof of (cOPEN x0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (f x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (f x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (f x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (f x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (f x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (f x0))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (f x0))
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b0)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((eq_trans0000 x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_trans000 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((((eq_trans0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((eq_trans0000 x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_trans000 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((((eq_trans0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((eq_trans0000 x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_trans000 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((((eq_trans0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) P0)) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((eq_trans0000 x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) b)) (x2:(((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_trans000 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((eq_trans00 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((((eq_trans0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0)) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0))) as proof of ((P0 (f x0))->(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x1:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (x2:(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((((eq_trans Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x1) x2) (fun (x4:Prop)=> ((P0 (f x0))->(P0 x4))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))) (((eq_ref Prop) (f x0)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found eq_ref000:=(eq_ref00 C):((C Xx)->(C Xx))
% Found (eq_ref00 C) as proof of ((C Xx)->(x0 Xx))
% Found ((eq_ref0 Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (((eq_ref fofType) Xx) C) as proof of ((C Xx)->(x0 Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType)=> (((eq_ref fofType) Xx) C)) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found x000:(C Xx)
% Instantiate: x0:=C:(fofType->Prop)
% Found (fun (x000:(C Xx))=> x000) as proof of (x0 Xx)
% Found (fun (Xx:fofType) (x000:(C Xx))=> x000) as proof of ((C Xx)->(x0 Xx))
% Found (fun (x00:((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))) (Xx:fofType) (x000:(C Xx))=> x000) as proof of (forall (Xx:fofType), ((C Xx)->(x0 Xx)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (b x)))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (b x0))
% Found (fun (x0:(fofType->Prop))=> ((eq_ref Prop) (f x0))) as proof of (forall (x:(fofType->Prop)), (((eq Prop) (f x)) (b x)))
% Found eta_expansion:=(fun (A:Type) (B:Type)=> ((eta_expansion_dep A) (fun (x1:A)=> B))):(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x))))
% Instantiate: b:=(forall (A:Type) (B:Type) (f:(A->B)), (((eq (A->B)) f) (fun (x:A)=> (f x)))):Prop
% Found eta_expansion as proof of b
% Found eta_expansion as proof of a
% Found eq_ref000:=(eq_ref00 x0):((x0 Xx)->(x0 Xx))
% Found (eq_ref00 x0) as proof of ((x0 Xx)->(A Xx))
% Found ((eq_ref0 Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (((eq_ref fofType) Xx) x0) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of ((x0 Xx)->(A Xx))
% Found (fun (Xx:fofType)=> (((eq_ref fofType) Xx) x0)) as proof of (forall (Xx:fofType), ((x0 Xx)->(A Xx)))
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) x0)
% Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) (fun (x:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x))))))
% Found (eta_expansion00 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq Prop) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq Prop) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found conj1:=(conj (cOPEN x0)):(forall (B:Prop), ((cOPEN x0)->(B->((and (cOPEN x0)) B))))
% Instantiate: b:=(forall (B:Prop), ((cOPEN x0)->(B->((and (cOPEN x0)) B)))):Prop
% Found conj1 as proof of b
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq Prop) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S x2)))))
% Found eq_ref00:=(eq_ref0 (x0 x2)):(((eq Prop) (x0 x2)) (x0 x2))
% Found (eq_ref0 (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found ((eq_ref Prop) (x0 x2)) as proof of (((eq Prop) (x0 x2)) b)
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found ((eq_ref Prop) b) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))):(((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Found (eq_ref0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) b)
% Found x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Instantiate: b:=((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_sym0100 ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found ((eq_sym0100 ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found (((fun (x2:(((eq Prop) (f x0)) b))=> ((eq_sym010 x2) P0)) ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((eq_sym01 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found (fun (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1)) as proof of (P0 (f x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1)) as proof of ((P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))->(P0 (f x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1)) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found (eq_sym000 (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found ((eq_sym00 (f x0)) (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found (((eq_sym0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found ((((eq_sym Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((eq_sym Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found ((((eq_sym Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((eq_sym Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Instantiate: b:=((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_sym0100 ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found ((eq_sym0100 ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found (((fun (x2:(((eq Prop) (f x0)) b))=> ((eq_sym010 x2) P0)) ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((eq_sym01 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1) as proof of (P0 (f x0))
% Found (fun (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1)) as proof of (P0 (f x0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1)) as proof of ((P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))->(P0 (f x0)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1)) as proof of (((eq Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0))
% Found (eq_sym000 (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found ((eq_sym00 (f x0)) (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found (((eq_sym0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> ((((eq_sym0 (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found ((((eq_sym Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((eq_sym Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found ((((eq_sym Prop) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) (f x0)) (fun (P0:(Prop->Prop)) (x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((fun (x2:(((eq Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))))=> (((((eq_sym Prop) (f x0)) ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))) x2) P0)) ((eq_ref Prop) (f x0))) x1))) as proof of (forall (P:(Prop->Prop)), ((P (f x0))->(P ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))))
% Found x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Instantiate: b:=((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Instantiate: b:=((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Instantiate: b:=((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found x1:(P0 ((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))))
% Instantiate: b:=((and ((and (cOPEN x0)) (forall (Xx:fofType), ((x0 Xx)->(A Xx))))) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx)))))):Prop
% Found x1 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (C:(fofType->Prop)), (((and (cOPEN C)) (forall (Xx:fofType), ((C Xx)->(A Xx))))->(forall (Xx:fofType), ((C Xx)->(x0 Xx))))))
% Found eq_ref000:=(eq_ref00 G):((G Xx)->(G Xx))
% Found (eq_ref00 G) as proof of ((G Xx)->(cOPEN Xx))
% Found ((eq_ref0 Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (((eq_ref (fofType->Prop)) Xx) G) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of ((G Xx)->(cOPEN Xx))
% Found (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G)) as proof of (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))
% Found ((conj20 (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) conj1) as proof of (P b)
% Found (((conj2 b) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) conj1) as proof of (P b)
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) b) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) conj1) as proof of (P b)
% Found ((((conj (forall (Xx:(fofType->Prop)), ((G Xx)->(cOPEN Xx)))) b) (fun (Xx:(fofType->Prop))=> (((eq_ref (fofType->Prop)) Xx) G))) conj1) as proof of (P b)
% Found eq_ref000:=(eq_ref00 P):((P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))->(P (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))))
% Found (eq_ref00 P) as proof of (P0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx))))))
% Found ((eq_ref0 (fun (Xx:fofType)=> ((ex (fofType->Prop)) (fun (S:(fofType->Prop))=> ((and (G S)) (S Xx)))))) P) as proof of (P0 (fun
% EOF
%------------------------------------------------------------------------------