%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV108^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n103.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 13:33:45 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  : SEV108^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n103.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 08:06:41 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 (forall (R:(fofType->(fofType->Prop))) (A:fofType) (B:fofType) (C:fofType) (D:fofType) (E:fofType) (F:fofType), (((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx:fofType) (Xy:fofType), (((R Xx) Xy)->((R Xy) Xx)))) (not (((eq fofType) A) B)))) (not (((eq fofType) A) C)))) (not (((eq fofType) A) D)))) (not (((eq fofType) A) E)))) (not (((eq fofType) A) F)))) (not (((eq fofType) B) C)))) (not (((eq fofType) B) D)))) (not (((eq fofType) B) E)))) (not (((eq fofType) B) F)))) (not (((eq fofType) C) D)))) (not (((eq fofType) C) E)))) (not (((eq fofType) C) F)))) (not (((eq fofType) D) E)))) (not (((eq fofType) D) F)))) (not (((eq fofType) E) F)))->((ex fofType) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))))) of role conjecture named cSIX_THEOREM_pme
% Conjecture to prove = (forall (R:(fofType->(fofType->Prop))) (A:fofType) (B:fofType) (C:fofType) (D:fofType) (E:fofType) (F:fofType), (((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx:fofType) (Xy:fofType), (((R Xx) Xy)->((R Xy) Xx)))) (not (((eq fofType) A) B)))) (not (((eq fofType) A) C)))) (not (((eq fofType) A) D)))) (not (((eq fofType) A) E)))) (not (((eq fofType) A) F)))) (not (((eq fofType) B) C)))) (not (((eq fofType) B) D)))) (not (((eq fofType) B) E)))) (not (((eq fofType) B) F)))) (not (((eq fofType) C) D)))) (not (((eq fofType) C) E)))) (not (((eq fofType) C) F)))) (not (((eq fofType) D) E)))) (not (((eq fofType) D) F)))) (not (((eq fofType) E) F)))->((ex fofType) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(forall (R:(fofType->(fofType->Prop))) (A:fofType) (B:fofType) (C:fofType) (D:fofType) (E:fofType) (F:fofType), (((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx:fofType) (Xy:fofType), (((R Xx) Xy)->((R Xy) Xx)))) (not (((eq fofType) A) B)))) (not (((eq fofType) A) C)))) (not (((eq fofType) A) D)))) (not (((eq fofType) A) E)))) (not (((eq fofType) A) F)))) (not (((eq fofType) B) C)))) (not (((eq fofType) B) D)))) (not (((eq fofType) B) E)))) (not (((eq fofType) B) F)))) (not (((eq fofType) C) D)))) (not (((eq fofType) C) E)))) (not (((eq fofType) C) F)))) (not (((eq fofType) D) E)))) (not (((eq fofType) D) F)))) (not (((eq fofType) E) F)))->((ex fofType) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))))']
% Parameter fofType:Type.
% Trying to prove (forall (R:(fofType->(fofType->Prop))) (A:fofType) (B:fofType) (C:fofType) (D:fofType) (E:fofType) (F:fofType), (((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx:fofType) (Xy:fofType), (((R Xx) Xy)->((R Xy) Xx)))) (not (((eq fofType) A) B)))) (not (((eq fofType) A) C)))) (not (((eq fofType) A) D)))) (not (((eq fofType) A) E)))) (not (((eq fofType) A) F)))) (not (((eq fofType) B) C)))) (not (((eq fofType) B) D)))) (not (((eq fofType) B) E)))) (not (((eq fofType) B) F)))) (not (((eq fofType) C) D)))) (not (((eq fofType) C) E)))) (not (((eq fofType) C) F)))) (not (((eq fofType) D) E)))) (not (((eq fofType) D) F)))) (not (((eq fofType) E) F)))->((ex fofType) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))):(((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x) Xb))) (not (((eq fofType) x) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x) Xb)) ((R x) Xc))) ((R Xb) Xc))) ((and ((and (((R x) Xb)->False)) (((R x) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found (eta_expansion_dep00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found eq_ref00:=(eq_ref0 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))):(((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found (eq_ref0 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))):(((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x) Xb))) (not (((eq fofType) x) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x) Xb)) ((R x) Xc))) ((R Xb) Xc))) ((and ((and (((R x) Xb)->False)) (((R x) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found (eta_expansion00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))):(((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x) Xb))) (not (((eq fofType) x) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x) Xb)) ((R x) Xc))) ((R Xb) Xc))) ((and ((and (((R x) Xb)->False)) (((R x) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found (eta_expansion_dep00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x7:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x7:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x7:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x7:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))):(((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x) Xb))) (not (((eq fofType) x) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x) Xb)) ((R x) Xc))) ((R Xb) Xc))) ((and ((and (((R x) Xb)->False)) (((R x) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found (eta_expansion_dep00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x9:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x9:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x9:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x9:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))):(((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x) Xb))) (not (((eq fofType) x) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x) Xb)) ((R x) Xc))) ((R Xb) Xc))) ((and ((and (((R x) Xb)->False)) (((R x) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found (eta_expansion00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% Found (eta_expansion_dep00 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eta_expansion_dep0 (fun (x1:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found (((eta_expansion_dep fofType) (fun (x1:fofType)=> Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% Found eq_ref00:=(eq_ref0 (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))):(((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))
% Found (eq_ref0 (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% 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 (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))):(((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))
% Found (eq_ref0 (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))):(((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x) Xb))) (not (((eq fofType) x) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x) Xb)) ((R x) Xc))) ((R Xb) Xc))) ((and ((and (((R x) Xb)->False)) (((R x) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found (eta_expansion_dep00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found ((eta_expansion_dep0 (fun (x13:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x13:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x13:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x13:fofType)=> Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))):(((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) x))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) x) Xc)))) ((or ((and ((and ((R x0) x)) ((R x0) Xc))) ((R x) Xc))) ((and ((and (((R x0) x)->False)) (((R x0) Xc)->False))) (((R x) Xc)->False))))))))
% Found (eta_expansion_dep00 (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found ((eta_expansion_dep0 (fun (x4:fofType)=> Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x4:fofType)=> Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% 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 (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) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found (((eq_trans000 ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((((eq_trans00 ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found (((((eq_trans0 (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% 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) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found (((eq_trans000 ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((((eq_trans00 ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found (((((eq_trans0 (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found eq_ref00:=(eq_ref0 (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x4) Xb))) (not (((eq fofType) x4) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x4) Xb)) ((R x4) Xc))) ((R Xb) Xc))) ((and ((and (((R x4) Xb)->False)) (((R x4) Xc)->False))) (((R Xb) Xc)->False)))))))):(((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x4) Xb))) (not (((eq fofType) x4) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x4) Xb)) ((R x4) Xc))) ((R Xb) Xc))) ((and ((and (((R x4) Xb)->False)) (((R x4) Xc)->False))) (((R Xb) Xc)->False)))))))) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x4) Xb))) (not (((eq fofType) x4) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x4) Xb)) ((R x4) Xc))) ((R Xb) Xc))) ((and ((and (((R x4) Xb)->False)) (((R x4) Xc)->False))) (((R Xb) Xc)->False))))))))
% Found (eq_ref0 (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x4) Xb))) (not (((eq fofType) x4) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x4) Xb)) ((R x4) Xc))) ((R Xb) Xc))) ((and ((and (((R x4) Xb)->False)) (((R x4) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x4) Xb))) (not (((eq fofType) x4) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x4) Xb)) ((R x4) Xc))) ((R Xb) Xc))) ((and ((and (((R x4) Xb)->False)) (((R x4) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x4) Xb))) (not (((eq fofType) x4) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x4) Xb)) ((R x4) Xc))) ((R Xb) Xc))) ((and ((and (((R x4) Xb)->False)) (((R x4) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x4) Xb))) (not (((eq fofType) x4) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x4) Xb)) ((R x4) Xc))) ((R Xb) Xc))) ((and ((and (((R x4) Xb)->False)) (((R x4) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x4) Xb))) (not (((eq fofType) x4) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x4) Xb)) ((R x4) Xc))) ((R Xb) Xc))) ((and ((and (((R x4) Xb)->False)) (((R x4) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x4) Xb))) (not (((eq fofType) x4) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x4) Xb)) ((R x4) Xc))) ((R Xb) Xc))) ((and ((and (((R x4) Xb)->False)) (((R x4) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found ((eq_ref (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x4) Xb))) (not (((eq fofType) x4) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x4) Xb)) ((R x4) Xc))) ((R Xb) Xc))) ((and ((and (((R x4) Xb)->False)) (((R x4) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x4) Xb))) (not (((eq fofType) x4) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x4) Xb)) ((R x4) Xc))) ((R Xb) Xc))) ((and ((and (((R x4) Xb)->False)) (((R x4) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% 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 eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))):(((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) x))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) x) Xc)))) ((or ((and ((and ((R x0) x)) ((R x0) Xc))) ((R x) Xc))) ((and ((and (((R x0) x)->False)) (((R x0) Xc)->False))) (((R x) Xc)->False))))))))
% Found (eta_expansion_dep00 (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found ((eta_expansion_dep0 (fun (x6:fofType)=> Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x0) Xb))) (not (((eq fofType) x0) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x0) Xb)) ((R x0) Xc))) ((R Xb) Xc))) ((and ((and (((R x0) Xb)->False)) (((R x0) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))):(((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) x))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) x) Xc)))) ((or ((and ((and ((R x2) x)) ((R x2) Xc))) ((R x) Xc))) ((and ((and (((R x2) x)->False)) (((R x2) Xc)->False))) (((R x) Xc)->False))))))))
% Found (eta_expansion_dep00 (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found ((eta_expansion_dep0 (fun (x6:fofType)=> Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False))))))))))
% 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 eta_expansion000:=(eta_expansion00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))):(((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x) Xb))) (not (((eq fofType) x) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x) Xb)) ((R x) Xc))) ((R Xb) Xc))) ((and ((and (((R x) Xb)->False)) (((R x) Xc)->False))) (((R Xb) Xc)->False))))))))))
% Found (eta_expansion00 (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found (((eta_expansion fofType) Prop) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xa:fofType)=> ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) Xa) Xb))) (not (((eq fofType) Xa) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R Xa) Xb)) ((R Xa) Xc))) ((R Xb) Xc))) ((and ((and (((R Xa) Xb)->False)) (((R Xa) Xc)->False))) (((R Xb) Xc)->False)))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found (((eq_trans000 ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((((eq_trans00 ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found (((((eq_trans0 (f x2)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((((((eq_trans Prop) (f x2)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found (((eq_trans000 ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((((eq_trans00 ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found (((((eq_trans0 (f x2)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False)))))))))
% Found ((((((eq_trans Prop) (f x2)) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) (((R Xb) Xc)->False))))))))) ((ex fofType) (fun (Xb:fofType)=> ((ex fofType) (fun (Xc:fofType)=> ((and ((and ((and (not (((eq fofType) x2) Xb))) (not (((eq fofType) x2) Xc)))) (not (((eq fofType) Xb) Xc)))) ((or ((and ((and ((R x2) Xb)) ((R x2) Xc))) ((R Xb) Xc))) ((and ((and (((R x2) Xb)->False)) (((R x2) Xc)->False))) ((
% EOF
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