%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SYO247^5 : TPTP v7.5.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox2/benchmark/theBenchmark.p

% Computer : n028.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 0s
% DateTime : Tue Mar 29 00:51:00 EDT 2022

% Result   : Timeout 300.03s 300.49s
% Output   : None 
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.03/0.11  % Problem    : SYO247^5 : TPTP v7.5.0. Released v4.0.0.
% 0.03/0.11  % Command    : python CASC.py /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.11/0.32  % Computer   : n028.cluster.edu
% 0.11/0.32  % Model      : x86_64 x86_64
% 0.11/0.32  % CPUModel   : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.11/0.32  % RAMPerCPU  : 8042.1875MB
% 0.11/0.32  % OS         : Linux 3.10.0-693.el7.x86_64
% 0.11/0.32  % CPULimit   : 300
% 0.11/0.32  % DateTime   : Fri Mar 11 21:34:18 EST 2022
% 0.11/0.32  % CPUTime    : 
% 0.11/0.33  ModuleCmd_Load.c(213):ERROR:105: Unable to locate a modulefile for 'python/python27'
% 0.11/0.34  Python 2.7.5
% 3.63/3.79  Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox2/benchmark/', '/export/starexec/sandbox2/benchmark/']
% 3.63/3.79  FOF formula (<kernel.Constant object at 0x2b4e06655098>, <kernel.Constant object at 0x1513710>) of role type named cP6
% 3.63/3.79  Using role type
% 3.63/3.79  Declaring cP6:fofType
% 3.63/3.79  FOF formula (<kernel.Constant object at 0x14eda28>, <kernel.Single object at 0x17aec68>) of role type named cP5
% 3.63/3.79  Using role type
% 3.63/3.79  Declaring cP5:fofType
% 3.63/3.79  FOF formula (<kernel.Constant object at 0x2b4e06655098>, <kernel.Single object at 0x15137e8>) of role type named cP4
% 3.63/3.79  Using role type
% 3.63/3.79  Declaring cP4:fofType
% 3.63/3.79  FOF formula (<kernel.Constant object at 0x17aec68>, <kernel.Single object at 0x1513758>) of role type named cP3
% 3.63/3.79  Using role type
% 3.63/3.79  Declaring cP3:fofType
% 3.63/3.79  FOF formula (<kernel.Constant object at 0x17aec68>, <kernel.Single object at 0x1513488>) of role type named cP2
% 3.63/3.79  Using role type
% 3.63/3.79  Declaring cP2:fofType
% 3.63/3.79  FOF formula (<kernel.Constant object at 0x15139e0>, <kernel.Single object at 0x1513e60>) of role type named cP1
% 3.63/3.79  Using role type
% 3.63/3.79  Declaring cP1:fofType
% 3.63/3.79  FOF formula (forall (K:(fofType->(fofType->Prop))) (S:(fofType->Prop)), (((and ((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))->((ex fofType) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))))) of role conjecture named cSIXPEOPLE2
% 3.63/3.79  Conjecture to prove = (forall (K:(fofType->(fofType->Prop))) (S:(fofType->Prop)), (((and ((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))->((ex fofType) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))))):Prop
% 3.63/3.79  We need to prove ['(forall (K:(fofType->(fofType->Prop))) (S:(fofType->Prop)), (((and ((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))->((ex fofType) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))))']
% 3.63/3.79  Parameter fofType:Type.
% 3.63/3.79  Parameter cP6:fofType.
% 3.63/3.79  Parameter cP5:fofType.
% 3.63/3.79  Parameter cP4:fofType.
% 3.63/3.79  Parameter cP3:fofType.
% 3.63/3.79  Parameter cP2:fofType.
% 3.63/3.79  Parameter cP1:fofType.
% 3.63/3.79  Trying to prove (forall (K:(fofType->(fofType->Prop))) (S:(fofType->Prop)), (((and ((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))->((ex fofType) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))))
% 3.63/3.79  Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 3.63/3.79  Found (eq_ref0 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 3.63/3.79  Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 3.63/3.79  Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 3.63/3.79  Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 4.84/5.04  Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x)) (S Xy))) (S Xz))) (not (((eq fofType) x) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x)))) ((or ((and ((and ((K x) Xy)) ((K Xy) Xz))) ((K x) Xz))) ((and ((and (((K x) Xy)->False)) (((K Xy) Xz)->False))) (((K x) Xz)->False))))))))))
% 4.84/5.04  Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 4.84/5.04  Found ((eta_expansion_dep0 (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 4.84/5.04  Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 4.84/5.04  Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 7.10/7.28  Found (((eta_expansion_dep fofType) (fun (x3:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 7.10/7.28  Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x)) (S Xy))) (S Xz))) (not (((eq fofType) x) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x)))) ((or ((and ((and ((K x) Xy)) ((K Xy) Xz))) ((K x) Xz))) ((and ((and (((K x) Xy)->False)) (((K Xy) Xz)->False))) (((K x) Xz)->False))))))))))
% 7.10/7.28  Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 7.10/7.28  Found ((eta_expansion_dep0 (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 11.73/11.91  Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 11.73/11.91  Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 11.73/11.91  Found (((eta_expansion_dep fofType) (fun (x5:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 11.73/11.91  Found eta_expansion000:=(eta_expansion00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x)) (S Xy))) (S Xz))) (not (((eq fofType) x) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x)))) ((or ((and ((and ((K x) Xy)) ((K Xy) Xz))) ((K x) Xz))) ((and ((and (((K x) Xy)->False)) (((K Xy) Xz)->False))) (((K x) Xz)->False))))))))))
% 11.73/11.92  Found (eta_expansion00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 11.73/11.92  Found ((eta_expansion0 Prop) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 11.73/11.92  Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 11.73/11.92  Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 11.73/11.92  Found (((eta_expansion fofType) Prop) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 19.01/19.20  Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x)) (S Xy))) (S Xz))) (not (((eq fofType) x) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x)))) ((or ((and ((and ((K x) Xy)) ((K Xy) Xz))) ((K x) Xz))) ((and ((and (((K x) Xy)->False)) (((K Xy) Xz)->False))) (((K x) Xz)->False))))))))))
% 19.01/19.20  Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 19.01/19.20  Found ((eta_expansion_dep0 (fun (x9:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 19.01/19.20  Found (((eta_expansion_dep fofType) (fun (x9:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 30.40/30.62  Found (((eta_expansion_dep fofType) (fun (x9:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 30.40/30.62  Found (((eta_expansion_dep fofType) (fun (x9:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 30.40/30.62  Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x)) (S Xy))) (S Xz))) (not (((eq fofType) x) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x)))) ((or ((and ((and ((K x) Xy)) ((K Xy) Xz))) ((K x) Xz))) ((and ((and (((K x) Xy)->False)) (((K Xy) Xz)->False))) (((K x) Xz)->False))))))))))
% 30.40/30.62  Found (eta_expansion_dep00 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 30.40/30.62  Found ((eta_expansion_dep0 (fun (x11:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 42.32/42.52  Found (((eta_expansion_dep fofType) (fun (x11:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 42.32/42.52  Found (((eta_expansion_dep fofType) (fun (x11:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 42.32/42.52  Found (((eta_expansion_dep fofType) (fun (x11:fofType)=> Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 42.32/42.52  Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 42.32/42.52  Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 42.32/42.52  Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 48.76/49.03  Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 48.76/49.03  Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 48.76/49.03  Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 48.76/49.03  Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% 48.76/49.03  Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 48.76/49.03  Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 48.76/49.03  Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 48.76/49.03  Found eq_ref00:=(eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))
% 48.76/49.03  Found (eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 48.76/49.03  Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 49.45/49.64  Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 49.45/49.64  Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 49.45/49.64  Found ex_intro000:=(ex_intro00 cP6):((S cP6)->((ex fofType) S))
% 49.45/49.64  Found (ex_intro00 cP6) as proof of ((S cP6)->((ex fofType) b))
% 49.45/49.64  Found ((ex_intro0 S) cP6) as proof of ((S cP6)->((ex fofType) b))
% 49.45/49.64  Found (((ex_intro fofType) S) cP6) as proof of ((S cP6)->((ex fofType) b))
% 49.45/49.64  Found (((ex_intro fofType) S) cP6) as proof of ((S cP6)->((ex fofType) b))
% 49.45/49.64  Found (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)) as proof of ((S cP6)->((ex fofType) b))
% 49.45/49.64  Found (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)) as proof of (((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->((ex fofType) b)))
% 49.45/49.64  Found (and_rect10 (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) b)
% 49.45/49.64  Found ((and_rect1 ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) b)
% 49.45/49.64  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) b)
% 49.45/49.64  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) b)
% 49.45/49.64  Found (fun (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)))) as proof of (P b)
% 49.45/49.64  Found (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)))) as proof of ((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->(P b))
% 49.49/49.68  Found (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)))) as proof of (((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->(P b)))
% 49.49/49.68  Found (and_rect00 (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))))) as proof of (P b)
% 49.49/49.68  Found ((and_rect0 (P b)) (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))))) as proof of (P b)
% 49.49/49.68  Found (((fun (P0:Type) (x0:(((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))) P0) x0) x)) (P b)) (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))))) as proof of (P b)
% 49.49/49.68  Found (((fun (P0:Type) (x0:(((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))) P0) x0) x)) (P b)) (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))))) as proof of (P b)
% 49.49/49.68  Found ex_intro000:=(ex_intro00 cP6):((S cP6)->((ex fofType) S))
% 53.34/53.53  Found (ex_intro00 cP6) as proof of ((S cP6)->((ex fofType) b))
% 53.34/53.53  Found ((ex_intro0 S) cP6) as proof of ((S cP6)->((ex fofType) b))
% 53.34/53.53  Found (((ex_intro fofType) S) cP6) as proof of ((S cP6)->((ex fofType) b))
% 53.34/53.53  Found (((ex_intro fofType) S) cP6) as proof of ((S cP6)->((ex fofType) b))
% 53.34/53.53  Found (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)) as proof of ((S cP6)->((ex fofType) b))
% 53.34/53.53  Found (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)) as proof of (((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->((ex fofType) b)))
% 53.34/53.53  Found (and_rect10 (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) b)
% 53.34/53.53  Found ((and_rect1 ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) b)
% 53.34/53.53  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) b)
% 53.34/53.53  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) b)
% 53.34/53.53  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) b)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of (P b)
% 53.34/53.53  Found eq_ref00:=(eq_ref0 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))):(((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 53.34/53.53  Found (eq_ref0 (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 54.29/54.49  Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 54.29/54.49  Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 54.29/54.49  Found ((eq_ref (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) as proof of (((eq (fofType->Prop)) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False)))))))))) b)
% 54.29/54.49  Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 54.29/54.49  Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 54.29/54.49  Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 54.29/54.49  Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 63.15/63.34  Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 63.15/63.34  Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 63.15/63.34  Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% 63.15/63.34  Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 63.15/63.34  Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 63.15/63.34  Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 63.15/63.34  Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 63.15/63.34  Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 63.15/63.34  Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% 63.15/63.34  Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 63.15/63.34  Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 63.15/63.34  Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 63.15/63.34  Found eq_ref00:=(eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))
% 63.15/63.34  Found (eq_ref0 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 63.15/63.34  Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 63.15/63.34  Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 68.53/68.74  Found ((eq_ref (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 68.53/68.74  Found eta_expansion000:=(eta_expansion00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S x))) (S Xz))) (not (((eq fofType) x0) x)))) (not (((eq fofType) x) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) x)) ((K x) Xz))) ((K x0) Xz))) ((and ((and (((K x0) x)->False)) (((K x) Xz)->False))) (((K x0) Xz)->False))))))))
% 68.53/68.74  Found (eta_expansion00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 68.53/68.74  Found ((eta_expansion0 Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 68.53/68.74  Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 69.58/69.78  Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 69.58/69.78  Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 69.58/69.78  Found ex_intro000:=(ex_intro00 cP5):((S cP5)->((ex fofType) S))
% 69.58/69.78  Found (ex_intro00 cP5) as proof of ((S cP5)->((ex fofType) b))
% 69.58/69.78  Found ((ex_intro0 S) cP5) as proof of ((S cP5)->((ex fofType) b))
% 69.58/69.78  Found (((ex_intro fofType) S) cP5) as proof of ((S cP5)->((ex fofType) b))
% 69.58/69.78  Found (((ex_intro fofType) S) cP5) as proof of ((S cP5)->((ex fofType) b))
% 69.58/69.78  Found (fun (x4:((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4)))=> (((ex_intro fofType) S) cP5)) as proof of ((S cP5)->((ex fofType) b))
% 69.58/69.78  Found (fun (x4:((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4)))=> (((ex_intro fofType) S) cP5)) as proof of (((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))->((S cP5)->((ex fofType) b)))
% 69.58/69.78  Found (and_rect20 (fun (x4:((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4)))=> (((ex_intro fofType) S) cP5))) as proof of ((ex fofType) b)
% 69.58/69.78  Found ((and_rect2 ((ex fofType) b)) (fun (x4:((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4)))=> (((ex_intro fofType) S) cP5))) as proof of ((ex fofType) b)
% 69.58/69.78  Found (((fun (P0:Type) (x4:(((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))->((S cP5)->P0)))=> (((((and_rect ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)) P0) x4) x2)) ((ex fofType) b)) (fun (x4:((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4)))=> (((ex_intro fofType) S) cP5))) as proof of ((ex fofType) b)
% 69.58/69.78  Found (((fun (P0:Type) (x4:(((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))->((S cP5)->P0)))=> (((((and_rect ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)) P0) x4) x2)) ((ex fofType) b)) (fun (x4:((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4)))=> (((ex_intro fofType) S) cP5))) as proof of ((ex fofType) b)
% 69.58/69.78  Found (((fun (P0:Type) (x4:(((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))->((S cP5)->P0)))=> (((((and_rect ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)) P0) x4) x2)) ((ex fofType) b)) (fun (x4:((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4)))=> (((ex_intro fofType) S) cP5))) as proof of (P b)
% 84.21/84.47  Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 84.21/84.47  Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 84.21/84.47  Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 84.21/84.47  Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 84.21/84.47  Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 84.21/84.47  Found eta_expansion000:=(eta_expansion00 a):(((eq (fofType->Prop)) a) (fun (x:fofType)=> (a x)))
% 84.21/84.47  Found (eta_expansion00 a) as proof of (((eq (fofType->Prop)) a) b)
% 84.21/84.47  Found ((eta_expansion0 Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 84.21/84.47  Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 84.21/84.47  Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 84.21/84.47  Found (((eta_expansion fofType) Prop) a) as proof of (((eq (fofType->Prop)) a) b)
% 84.21/84.47  Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 84.21/84.47  Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% 84.21/84.47  Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 84.21/84.47  Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 84.21/84.47  Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 84.21/84.47  Found ex_intro000:=(ex_intro00 cP6):((S cP6)->((ex fofType) S))
% 84.21/84.47  Found (ex_intro00 cP6) as proof of ((S cP6)->((ex fofType) f))
% 84.21/84.47  Found ((ex_intro0 S) cP6) as proof of ((S cP6)->((ex fofType) f))
% 84.21/84.47  Found (((ex_intro fofType) S) cP6) as proof of ((S cP6)->((ex fofType) f))
% 84.21/84.47  Found (((ex_intro fofType) S) cP6) as proof of ((S cP6)->((ex fofType) f))
% 84.21/84.47  Found (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)) as proof of ((S cP6)->((ex fofType) f))
% 84.21/84.47  Found (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)) as proof of (((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->((ex fofType) f)))
% 84.21/84.47  Found (and_rect10 (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 84.21/84.47  Found ((and_rect1 ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 84.21/84.47  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 114.59/114.83  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 114.59/114.83  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of (P f)
% 114.59/114.83  Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 114.59/114.83  Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 114.59/114.83  Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 114.59/114.83  Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 114.59/114.83  Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 114.59/114.83  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 114.59/114.83  Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 114.59/114.83  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 114.59/114.83  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 114.59/114.83  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 114.59/114.83  Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 114.59/114.83  Found (((eq_trans000 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 114.59/114.83  Found ((((eq_trans00 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 114.59/114.83  Found (((((eq_trans0 (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 114.59/114.83  Found ((((((eq_trans Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 120.41/120.66  Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 120.41/120.66  Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 120.41/120.66  Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 120.41/120.66  Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 120.41/120.66  Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 120.41/120.66  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 120.41/120.66  Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 120.41/120.66  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 120.41/120.66  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 120.41/120.66  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 120.41/120.66  Found x30:(P0 (f x2))
% 120.41/120.66  Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% 120.41/120.66  Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% 120.41/120.66  Found x30:(P0 (f x2))
% 120.41/120.66  Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% 120.41/120.66  Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% 120.41/120.66  Found ex_intro000:=(ex_intro00 cP6):((S cP6)->((ex fofType) S))
% 120.41/120.66  Found (ex_intro00 cP6) as proof of ((S cP6)->((ex fofType) f))
% 120.41/120.66  Found ((ex_intro0 S) cP6) as proof of ((S cP6)->((ex fofType) f))
% 120.41/120.66  Found (((ex_intro fofType) S) cP6) as proof of ((S cP6)->((ex fofType) f))
% 120.41/120.66  Found (((ex_intro fofType) S) cP6) as proof of ((S cP6)->((ex fofType) f))
% 120.41/120.66  Found (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)) as proof of ((S cP6)->((ex fofType) f))
% 120.41/120.66  Found (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)) as proof of (((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->((ex fofType) f)))
% 120.41/120.66  Found (and_rect10 (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 150.42/150.64  Found ((and_rect1 ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 150.42/150.64  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 150.42/150.64  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 150.42/150.64  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of (P f)
% 150.42/150.64  Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 150.42/150.64  Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 150.42/150.64  Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 150.42/150.64  Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 150.42/150.64  Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 150.42/150.64  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 150.42/150.64  Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 150.42/150.64  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 150.42/150.64  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 150.42/150.64  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 150.42/150.64  Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 150.42/150.64  Found (((eq_trans000 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 150.42/150.64  Found ((((eq_trans00 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 150.42/150.65  Found (((((eq_trans0 (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 150.42/150.65  Found ((((((eq_trans Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))) as proof of (((eq Prop) (f x2)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 158.92/159.23  Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% 158.92/159.23  Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% 158.92/159.23  Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 158.92/159.23  Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 158.92/159.23  Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% 158.92/159.23  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 158.92/159.23  Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 158.92/159.23  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 158.92/159.23  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 158.92/159.23  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 158.92/159.23  Found x30:(P0 (f x2))
% 158.92/159.23  Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% 158.92/159.23  Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% 158.92/159.23  Found x30:(P0 (f x2))
% 158.92/159.23  Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P0 (f x2))
% 158.92/159.23  Found (fun (x30:(P0 (f x2)))=> x30) as proof of (P1 (f x2))
% 158.92/159.23  Found ex_intro010:=(ex_intro01 cP6):((S cP6)->((ex fofType) S))
% 158.92/159.23  Found (ex_intro01 cP6) as proof of ((S cP6)->(P b))
% 158.92/159.23  Found ((ex_intro0 S) cP6) as proof of ((S cP6)->(P b))
% 158.92/159.23  Found ((ex_intro0 S) cP6) as proof of ((S cP6)->(P b))
% 158.92/159.23  Found (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6)) as proof of ((S cP6)->(P b))
% 158.92/159.23  Found (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6)) as proof of (((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->(P b)))
% 158.92/159.23  Found (and_rect10 (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))) as proof of (P b)
% 159.04/159.29  Found ((and_rect1 (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))) as proof of (P b)
% 159.04/159.29  Found (((fun (P0:Type) (x3:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))) as proof of (P b)
% 159.04/159.29  Found (((fun (P0:Type) (x3:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))) as proof of (P b)
% 159.04/159.29  Found ex_intro010:=(ex_intro01 cP6):((S cP6)->((ex fofType) S))
% 159.04/159.29  Found (ex_intro01 cP6) as proof of ((S cP6)->(P b))
% 159.04/159.29  Found ((ex_intro0 S) cP6) as proof of ((S cP6)->(P b))
% 159.04/159.29  Found ((ex_intro0 S) cP6) as proof of ((S cP6)->(P b))
% 159.04/159.29  Found (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6)) as proof of ((S cP6)->(P b))
% 159.04/159.29  Found (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6)) as proof of (((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->(P b)))
% 159.04/159.29  Found (and_rect10 (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))) as proof of (P b)
% 159.04/159.29  Found ((and_rect1 (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))) as proof of (P b)
% 159.04/159.29  Found (((fun (P0:Type) (x3:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x3) x0)) (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))) as proof of (P b)
% 159.04/159.29  Found (((fun (P0:Type) (x3:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x3) x0)) (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))) as proof of (P b)
% 159.04/159.29  Found ex_intro010:=(ex_intro01 cP6):((S cP6)->((ex fofType) S))
% 159.04/159.29  Found (ex_intro01 cP6) as proof of ((S cP6)->(P b))
% 159.04/159.29  Found ((ex_intro0 S) cP6) as proof of ((S cP6)->(P b))
% 159.04/159.29  Found ((ex_intro0 S) cP6) as proof of ((S cP6)->(P b))
% 159.04/159.29  Found (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6)) as proof of ((S cP6)->(P b))
% 159.04/159.29  Found (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6)) as proof of (((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->(P b)))
% 159.04/159.29  Found (and_rect10 (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))) as proof of (P b)
% 159.04/159.29  Found ((and_rect1 (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))) as proof of (P b)
% 159.04/159.29  Found (((fun (P0:Type) (x3:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))) as proof of (P b)
% 159.04/159.29  Found (fun (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6)))) as proof of (P b)
% 159.04/159.29  Found (fun (x1:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6)))) as proof of ((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->(P b))
% 159.04/159.29  Found (fun (x1:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6)))) as proof of (((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->(P b)))
% 159.04/159.29  Found (and_rect00 (fun (x1:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))))) as proof of (P b)
% 159.04/159.29  Found ((and_rect0 (P b)) (fun (x1:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))))) as proof of (P b)
% 159.04/159.29  Found (((fun (P0:Type) (x1:(((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))) P0) x1) x)) (P b)) (fun (x1:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))))) as proof of (P b)
% 159.04/159.29  Found (((fun (P0:Type) (x1:(((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))) P0) x1) x)) (P b)) (fun (x1:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x3:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x3) x1)) (P b)) (fun (x3:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> ((ex_intro0 S) cP6))))) as proof of (P b)
% 161.23/161.51  Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S x))) (S Xz))) (not (((eq fofType) x4) x)))) (not (((eq fofType) x) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) x)) ((K x) Xz))) ((K x4) Xz))) ((and ((and (((K x4) x)->False)) (((K x) Xz)->False))) (((K x4) Xz)->False))))))))
% 161.23/161.51  Found (eta_expansion_dep00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 161.23/161.51  Found ((eta_expansion_dep0 (fun (x6:fofType)=> Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 161.23/161.51  Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 161.23/161.51  Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 167.85/168.11  Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x4)) (S Xy))) (S Xz))) (not (((eq fofType) x4) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x4)))) ((or ((and ((and ((K x4) Xy)) ((K Xy) Xz))) ((K x4) Xz))) ((and ((and (((K x4) Xy)->False)) (((K Xy) Xz)->False))) (((K x4) Xz)->False)))))))) b)
% 167.85/168.11  Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 167.85/168.11  Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 167.85/168.11  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 167.85/168.11  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 167.85/168.11  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 167.85/168.11  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 167.85/168.11  Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 167.85/168.11  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 167.85/168.11  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 167.85/168.11  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 167.85/168.11  Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 167.85/168.11  Found (((eq_trans000 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 167.85/168.12  Found ((((eq_trans00 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 167.85/168.12  Found (((((eq_trans0 (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 167.85/168.12  Found ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 168.02/168.30  Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 168.02/168.30  Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 168.02/168.30  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 168.02/168.30  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 168.02/168.30  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 168.02/168.30  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 168.02/168.30  Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 168.02/168.30  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 168.02/168.30  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 168.02/168.30  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 168.02/168.30  Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 168.02/168.30  Found (((eq_trans000 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 168.02/168.31  Found ((((eq_trans00 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 168.02/168.31  Found (((((eq_trans0 (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 168.02/168.31  Found ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 175.06/175.31  Found eta_expansion000:=(eta_expansion00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S x))) (S Xz))) (not (((eq fofType) x2) x)))) (not (((eq fofType) x) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) x)) ((K x) Xz))) ((K x2) Xz))) ((and ((and (((K x2) x)->False)) (((K x) Xz)->False))) (((K x2) Xz)->False))))))))
% 175.06/175.31  Found (eta_expansion00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 175.06/175.31  Found ((eta_expansion0 Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 175.06/175.31  Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 175.11/175.44  Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 175.11/175.44  Found (((eta_expansion fofType) Prop) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))) b)
% 175.11/175.44  Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))):(((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) (fun (x:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S x))) (S Xz))) (not (((eq fofType) x0) x)))) (not (((eq fofType) x) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) x)) ((K x) Xz))) ((K x0) Xz))) ((and ((and (((K x0) x)->False)) (((K x) Xz)->False))) (((K x0) Xz)->False))))))))
% 175.11/175.44  Found (eta_expansion_dep00 (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 175.11/175.44  Found ((eta_expansion_dep0 (fun (x6:fofType)=> Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 188.28/188.55  Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 188.28/188.55  Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 188.28/188.55  Found (((eta_expansion_dep fofType) (fun (x6:fofType)=> Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) as proof of (((eq (fofType->Prop)) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))) b)
% 188.28/188.55  Found eq_ref00:=(eq_ref0 b):(((eq (fofType->Prop)) b) b)
% 188.28/188.55  Found (eq_ref0 b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 188.28/188.55  Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 188.28/188.55  Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 202.78/203.05  Found ((eq_ref (fofType->Prop)) b) as proof of (((eq (fofType->Prop)) b) (fun (Xx:fofType)=> ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S Xx)) (S Xy))) (S Xz))) (not (((eq fofType) Xx) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) Xx)))) ((or ((and ((and ((K Xx) Xy)) ((K Xy) Xz))) ((K Xx) Xz))) ((and ((and (((K Xx) Xy)->False)) (((K Xy) Xz)->False))) (((K Xx) Xz)->False))))))))))
% 202.78/203.05  Found eq_ref00:=(eq_ref0 a):(((eq (fofType->Prop)) a) a)
% 202.78/203.05  Found (eq_ref0 a) as proof of (((eq (fofType->Prop)) a) b)
% 202.78/203.05  Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 202.78/203.05  Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 202.78/203.05  Found ((eq_ref (fofType->Prop)) a) as proof of (((eq (fofType->Prop)) a) b)
% 202.78/203.05  Found ex_intro000:=(ex_intro00 cP4):((S cP4)->((ex fofType) S))
% 202.78/203.05  Found (ex_intro00 cP4) as proof of ((S cP4)->((ex fofType) b))
% 202.78/203.05  Found ((ex_intro0 S) cP4) as proof of ((S cP4)->((ex fofType) b))
% 202.78/203.05  Found (((ex_intro fofType) S) cP4) as proof of ((S cP4)->((ex fofType) b))
% 202.78/203.05  Found (((ex_intro fofType) S) cP4) as proof of ((S cP4)->((ex fofType) b))
% 202.78/203.05  Found (fun (x6:((and ((and (S cP1)) (S cP2))) (S cP3)))=> (((ex_intro fofType) S) cP4)) as proof of ((S cP4)->((ex fofType) b))
% 202.78/203.05  Found (fun (x6:((and ((and (S cP1)) (S cP2))) (S cP3)))=> (((ex_intro fofType) S) cP4)) as proof of (((and ((and (S cP1)) (S cP2))) (S cP3))->((S cP4)->((ex fofType) b)))
% 202.78/203.05  Found (and_rect30 (fun (x6:((and ((and (S cP1)) (S cP2))) (S cP3)))=> (((ex_intro fofType) S) cP4))) as proof of ((ex fofType) b)
% 202.78/203.05  Found ((and_rect3 ((ex fofType) b)) (fun (x6:((and ((and (S cP1)) (S cP2))) (S cP3)))=> (((ex_intro fofType) S) cP4))) as proof of ((ex fofType) b)
% 202.78/203.05  Found (((fun (P0:Type) (x6:(((and ((and (S cP1)) (S cP2))) (S cP3))->((S cP4)->P0)))=> (((((and_rect ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4)) P0) x6) x4)) ((ex fofType) b)) (fun (x6:((and ((and (S cP1)) (S cP2))) (S cP3)))=> (((ex_intro fofType) S) cP4))) as proof of ((ex fofType) b)
% 202.78/203.05  Found (((fun (P0:Type) (x6:(((and ((and (S cP1)) (S cP2))) (S cP3))->((S cP4)->P0)))=> (((((and_rect ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4)) P0) x6) x4)) ((ex fofType) b)) (fun (x6:((and ((and (S cP1)) (S cP2))) (S cP3)))=> (((ex_intro fofType) S) cP4))) as proof of ((ex fofType) b)
% 202.78/203.05  Found (((fun (P0:Type) (x6:(((and ((and (S cP1)) (S cP2))) (S cP3))->((S cP4)->P0)))=> (((((and_rect ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4)) P0) x6) x4)) ((ex fofType) b)) (fun (x6:((and ((and (S cP1)) (S cP2))) (S cP3)))=> (((ex_intro fofType) S) cP4))) as proof of (P b)
% 202.78/203.05  Found x10:(P0 (f x0))
% 202.78/203.05  Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% 202.78/203.05  Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% 202.78/203.05  Found x10:(P0 (f x0))
% 202.78/203.05  Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% 202.78/203.05  Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% 202.78/203.05  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 202.78/203.05  Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% 202.78/203.05  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 202.78/203.05  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 202.78/203.05  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 202.78/203.05  Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))):(((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 203.19/203.53  Found (eq_ref0 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) b)
% 203.19/203.53  Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) b)
% 203.19/203.53  Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) b)
% 203.19/203.53  Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) b)
% 203.19/203.53  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 203.19/203.53  Found (eq_ref0 b) as proof of (((eq Prop) b) (f x2))
% 203.19/203.53  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 203.19/203.53  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 203.19/203.53  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x2))
% 203.19/203.53  Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))):(((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False)))))))))
% 205.06/205.39  Found (eq_ref0 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) b)
% 205.06/205.39  Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) b)
% 205.06/205.39  Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) b)
% 205.06/205.39  Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x2)) (S Xy))) (S Xz))) (not (((eq fofType) x2) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x2)))) ((or ((and ((and ((K x2) Xy)) ((K Xy) Xz))) ((K x2) Xz))) ((and ((and (((K x2) Xy)->False)) (((K Xy) Xz)->False))) (((K x2) Xz)->False))))))))) b)
% 205.06/205.39  Found ex_intro000:=(ex_intro00 cP6):((S cP6)->((ex fofType) S))
% 205.06/205.39  Found (ex_intro00 cP6) as proof of ((S cP6)->((ex fofType) f))
% 205.06/205.39  Found ((ex_intro0 S) cP6) as proof of ((S cP6)->((ex fofType) f))
% 205.06/205.39  Found (((ex_intro fofType) S) cP6) as proof of ((S cP6)->((ex fofType) f))
% 205.06/205.39  Found (((ex_intro fofType) S) cP6) as proof of ((S cP6)->((ex fofType) f))
% 205.06/205.39  Found (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)) as proof of ((S cP6)->((ex fofType) f))
% 205.06/205.39  Found (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)) as proof of (((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->((ex fofType) f)))
% 205.06/205.39  Found (and_rect10 (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 205.06/205.39  Found ((and_rect1 ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 205.06/205.39  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 205.06/205.39  Found (fun (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)))) as proof of ((ex fofType) f)
% 205.06/205.39  Found (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)))) as proof of ((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->((ex fofType) f))
% 205.06/205.39  Found (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)))) as proof of (((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->((ex fofType) f)))
% 205.06/205.39  Found (and_rect00 (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))))) as proof of ((ex fofType) f)
% 205.06/205.39  Found ((and_rect0 ((ex fofType) f)) (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))))) as proof of ((ex fofType) f)
% 234.88/235.16  Found (((fun (P0:Type) (x0:(((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))) P0) x0) x)) ((ex fofType) f)) (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))))) as proof of ((ex fofType) f)
% 234.88/235.16  Found (((fun (P0:Type) (x0:(((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))) P0) x0) x)) ((ex fofType) f)) (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))))) as proof of ((ex fofType) f)
% 234.88/235.16  Found (((fun (P0:Type) (x0:(((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))) P0) x0) x)) ((ex fofType) f)) (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))))) as proof of (P f)
% 234.88/235.16  Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 234.88/235.16  Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 234.88/235.16  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 234.88/235.16  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 234.88/235.16  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 234.88/235.16  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 234.88/235.16  Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 234.88/235.16  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 256.96/257.35  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 256.96/257.35  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 256.96/257.35  Found x10:(P0 (f x0))
% 256.96/257.35  Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P0 (f x0))
% 256.96/257.35  Found (fun (x10:(P0 (f x0)))=> x10) as proof of (P1 (f x0))
% 256.96/257.35  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 256.96/257.35  Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 256.96/257.35  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 256.96/257.35  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 256.96/257.35  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 256.96/257.35  Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))):(((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 256.96/257.35  Found (eq_ref0 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 256.96/257.35  Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 256.96/257.35  Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 259.51/259.84  Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 259.51/259.84  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 259.51/259.84  Found (eq_ref0 b) as proof of (((eq Prop) b) (f x0))
% 259.51/259.84  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 259.51/259.84  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 259.51/259.84  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x0))
% 259.51/259.84  Found eq_ref00:=(eq_ref0 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))):(((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 259.51/259.84  Found (eq_ref0 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 259.51/259.84  Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 262.64/262.96  Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 262.64/262.96  Found ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) as proof of (((eq Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) b)
% 262.64/262.96  Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 262.64/262.96  Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 262.64/262.96  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 262.64/262.96  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 262.64/262.96  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 262.64/262.96  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 262.64/262.96  Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 262.64/262.96  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 262.64/262.96  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 262.64/262.96  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 262.64/262.96  Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 262.64/262.97  Found (((eq_trans000 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 262.64/262.97  Found ((((eq_trans00 ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 262.64/262.97  Found (((((eq_trans0 (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 262.64/262.97  Found ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 262.64/262.98  Found (fun (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 262.64/262.98  Found (fun (x1:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))))) as proof of ((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->(((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))))
% 262.64/263.00  Found (fun (x1:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))))) as proof of (((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->(((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))))
% 262.64/263.00  Found (and_rect00 (fun (x1:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 262.64/263.01  Found ((and_rect0 (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) (fun (x1:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 262.64/263.01  Found (((fun (P0:Type) (x1:(((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->P0)))=> (((((and_rect ((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))) P0) x1) x)) (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))) (fun (x1:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x2:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> ((((((eq_trans Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False))))))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))))) as proof of (((eq Prop) (f x0)) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 278.75/279.14  Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 278.75/279.14  Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% 278.75/279.14  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 278.75/279.14  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 278.75/279.14  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% 278.75/279.14  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 278.75/279.14  Found (eq_ref0 b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 278.75/279.14  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 278.75/279.14  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 278.75/279.14  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((ex fofType) (fun (Xy:fofType)=> ((ex fofType) (fun (Xz:fofType)=> ((and ((and ((and ((and ((and ((and (S x0)) (S Xy))) (S Xz))) (not (((eq fofType) x0) Xy)))) (not (((eq fofType) Xy) Xz)))) (not (((eq fofType) Xz) x0)))) ((or ((and ((and ((K x0) Xy)) ((K Xy) Xz))) ((K x0) Xz))) ((and ((and (((K x0) Xy)->False)) (((K Xy) Xz)->False))) (((K x0) Xz)->False)))))))))
% 278.75/279.14  Found x30:(P0 (f x0))
% 278.75/279.14  Found (fun (x30:(P0 (f x0)))=> x30) as proof of (P0 (f x0))
% 278.75/279.14  Found (fun (x30:(P0 (f x0)))=> x30) as proof of (P1 (f x0))
% 278.75/279.14  Found ex_intro000:=(ex_intro00 cP6):((S cP6)->((ex fofType) S))
% 278.75/279.14  Found (ex_intro00 cP6) as proof of ((S cP6)->((ex fofType) f))
% 278.75/279.14  Found ((ex_intro0 S) cP6) as proof of ((S cP6)->((ex fofType) f))
% 278.75/279.14  Found (((ex_intro fofType) S) cP6) as proof of ((S cP6)->((ex fofType) f))
% 278.75/279.14  Found (((ex_intro fofType) S) cP6) as proof of ((S cP6)->((ex fofType) f))
% 278.75/279.14  Found (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)) as proof of ((S cP6)->((ex fofType) f))
% 278.75/279.14  Found (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)) as proof of (((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->((ex fofType) f)))
% 278.75/279.14  Found (and_rect10 (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 278.75/279.15  Found ((and_rect1 ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 278.75/279.15  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 278.75/279.15  Found (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))) as proof of ((ex fofType) f)
% 278.75/279.15  Found (fun (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)))) as proof of (P f)
% 278.75/279.15  Found (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)))) as proof of ((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->(P f))
% 278.75/279.15  Found (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6)))) as proof of (((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))->((forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx)))->(P f)))
% 278.75/279.15  Found (and_rect00 (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))))) as proof of (P f)
% 278.75/279.15  Found ((and_rect0 (P f)) (fun (x0:((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6))) (x1:(forall (Xx:fofType) (Xy:fofType), (((K Xx) Xy)->((K Xy) Xx))))=> (((fun (P0:Type) (x2:(((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))->((S cP6)->P0)))=> (((((and_rect ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5))) (S cP6)) P0) x2) x0)) ((ex fofType) f)) (fun (x2:((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4))) (S cP5)))=> (((ex_intro fofType) S) cP6))))) as proof of (P f)
% 278.75/279.15  Found (((fun (P0:Type) (x0:(((and ((and ((and ((and ((and (S cP1)) (S cP2))) (S cP3))) (S cP4)
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