TSTP Solution File: SEV129^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV129^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n099.star.cs.uiowa.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory : 32286.75MB
% OS : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:46 EDT 2014
% Result : Timeout 300.04s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV129^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n099.star.cs.uiowa.edu
% % Model : x86_64 x86_64
% % CPU : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory : 32286.75MB
% % OS : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:11:06 CDT 2014
% % CPUTime : 300.04
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x8dcd88>, <kernel.Type object at 0xd14a70>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (S:((a->(a->Prop))->Prop)) (Xx:a) (Xy:a), ((forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))))) of role conjecture named cTHM253_A_pme
% Conjecture to prove = (forall (S:((a->(a->Prop))->Prop)) (Xx:a) (Xy:a), ((forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (S:((a->(a->Prop))->Prop)) (Xx:a) (Xy:a), ((forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))))']
% Parameter a:Type.
% Trying to prove (forall (S:((a->(a->Prop))->Prop)) (Xx:a) (Xy:a), ((forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))))
% Found x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x2 as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x3 as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp10 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp10 Xx00) Xy00)) ((Xp10 Xy00) Xz))->((Xp10 Xx00) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp10 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp10 Xx0) Xy00)) ((Xp10 Xy00) Xz0))->((Xp10 Xx0) Xz0))))->((Xp10 Xy0) Xz))))->(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp10 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp10 Xx00) Xy0)) ((Xp10 Xy0) Xz0))->((Xp10 Xx00) Xz0))))->((Xp10 Xx0) Xz)))))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp10 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp10 Xx00) Xy00)) ((Xp10 Xy00) Xz))->((Xp10 Xx00) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp10 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp10 Xx0) Xy00)) ((Xp10 Xy00) Xz0))->((Xp10 Xx0) Xz0))))->((Xp10 Xy0) Xz))))->(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp10 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp10 Xx00) Xy0)) ((Xp10 Xy0) Xz0))->((Xp10 Xx00) Xz0))))->((Xp10 Xx0) Xz)))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp10 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp10 Xx00) Xy00)) ((Xp10 Xy00) Xz))->((Xp10 Xx00) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp10 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp10 Xx0) Xy00)) ((Xp10 Xy00) Xz0))->((Xp10 Xx0) Xz0))))->((Xp10 Xy0) Xz))))->(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp10 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp10 Xx00) Xy0)) ((Xp10 Xy0) Xz0))->((Xp10 Xx00) Xz0))))->((Xp10 Xx0) Xz))))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp10 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp10 Xx00) Xy00)) ((Xp10 Xy00) Xz))->((Xp10 Xx00) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp10 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp10 Xx0) Xy00)) ((Xp10 Xy00) Xz0))->((Xp10 Xx0) Xz0))))->((Xp10 Xy0) Xz))))->(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp10 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp10 Xx00) Xy0)) ((Xp10 Xy0) Xz0))->((Xp10 Xx00) Xz0))))->((Xp10 Xx0) Xz)))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp10 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp10 Xx00) Xy00)) ((Xp10 Xy00) Xz))->((Xp10 Xx00) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp10 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp10 Xx0) Xy00)) ((Xp10 Xy00) Xz0))->((Xp10 Xx0) Xz0))))->((Xp10 Xy0) Xz))))->(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp10 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp10 Xx00) Xy0)) ((Xp10 Xy0) Xz0))->((Xp10 Xx00) Xz0))))->((Xp10 Xx0) Xz)))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp10 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp10 Xx00) Xy00)) ((Xp10 Xy00) Xz))->((Xp10 Xx00) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp10 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp10 Xx0) Xy00)) ((Xp10 Xy00) Xz0))->((Xp10 Xx0) Xz0))))->((Xp10 Xy0) Xz))))->(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp10 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp10 Xx00) Xy0)) ((Xp10 Xy0) Xz0))->((Xp10 Xx00) Xz0))))->((Xp10 Xx0) Xz)))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp10 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp10 Xx00) Xy00)) ((Xp10 Xy00) Xz))->((Xp10 Xx00) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp10 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp10 Xx0) Xy00)) ((Xp10 Xy00) Xz0))->((Xp10 Xx0) Xz0))))->((Xp10 Xy0) Xz))))->(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp10 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp10 Xx00) Xy0)) ((Xp10 Xy0) Xz0))->((Xp10 Xx00) Xz0))))->((Xp10 Xx0) Xz)))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp10 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp10 Xx00) Xy00)) ((Xp10 Xy00) Xz))->((Xp10 Xx00) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp10 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp10 Xx0) Xy00)) ((Xp10 Xy00) Xz0))->((Xp10 Xx0) Xz0))))->((Xp10 Xy0) Xz))))->(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp10 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp10 Xx00) Xy0)) ((Xp10 Xy0) Xz0))->((Xp10 Xx00) Xz0))))->((Xp10 Xx0) Xz)))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp10 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp10 Xx00) Xy00)) ((Xp10 Xy00) Xz))->((Xp10 Xx00) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp10 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp10 Xx0) Xy00)) ((Xp10 Xy00) Xz0))->((Xp10 Xx0) Xz0))))->((Xp10 Xy0) Xz))))->(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp10 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp10 Xx00) Xy0)) ((Xp10 Xy0) Xz0))->((Xp10 Xx00) Xz0))))->((Xp10 Xx0) Xz)))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp10 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp10 Xx00) Xy00)) ((Xp10 Xy00) Xz))->((Xp10 Xx00) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp10 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp10 Xx0) Xy00)) ((Xp10 Xy00) Xz0))->((Xp10 Xx0) Xz0))))->((Xp10 Xy0) Xz))))->(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp10 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp10 Xx00) Xy0)) ((Xp10 Xy0) Xz0))->((Xp10 Xx00) Xz0))))->((Xp10 Xx0) Xz)))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp10 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp10 Xx00) Xy00)) ((Xp10 Xy00) Xz))->((Xp10 Xx00) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp10 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp10 Xx0) Xy00)) ((Xp10 Xy00) Xz0))->((Xp10 Xx0) Xz0))))->((Xp10 Xy0) Xz))))->(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp10 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp10 Xx00) Xy0)) ((Xp10 Xy0) Xz0))->((Xp10 Xx00) Xz0))))->((Xp10 Xx0) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz))))->((Xp1 Xx0) Xy0))) (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0))))->((Xp1 Xy0) Xz)))->(((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))->((Xp1 Xx0) Xz))))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz))))->((Xp1 Xx0) Xy0))) (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0))))->((Xp1 Xy0) Xz)))->(((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))->((Xp1 Xx0) Xz))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz))))->((Xp1 Xx0) Xy0))) (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0))))->((Xp1 Xy0) Xz)))->(((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))->((Xp1 Xx0) Xz)))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz))))->((Xp1 Xx0) Xy0))) (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0))))->((Xp1 Xy0) Xz)))->(((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))->((Xp1 Xx0) Xz))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz))))->((Xp1 Xx0) Xy0))) (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0))))->((Xp1 Xy0) Xz)))->(((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))->((Xp1 Xx0) Xz))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz))))->((Xp1 Xx0) Xy0))) (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0))))->((Xp1 Xy0) Xz)))->(((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))->((Xp1 Xx0) Xz))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz))))->((Xp1 Xx0) Xy0))) (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0))))->((Xp1 Xy0) Xz)))->(((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))->((Xp1 Xx0) Xz))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz))))->((Xp1 Xx0) Xy0))) (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0))))->((Xp1 Xy0) Xz)))->(((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))->((Xp1 Xx0) Xz))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz))))->((Xp1 Xx0) Xy0))) (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0))))->((Xp1 Xy0) Xz)))->(((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))->((Xp1 Xx0) Xz))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz))))->((Xp1 Xx0) Xy0))) (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0))))->((Xp1 Xy0) Xz)))->(((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))->((Xp1 Xx0) Xz))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and (((and (forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))) (forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz))))->((Xp1 Xx0) Xy0))) (((and (forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))) (forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0))))->((Xp1 Xy0) Xz)))->(((and (forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))->((Xp1 Xx0) Xz))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) b)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found eq_ref00:=(eq_ref0 Xy):(((eq a) Xy) Xy)
% Found (eq_ref0 Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found ((eq_ref a) Xy) as proof of (((eq a) Xy) b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))) ((forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))->((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz))))->((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz)))))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))) ((forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))->((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz))))->((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz)))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))) ((forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))->((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz))))->((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))) ((forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))->((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz))))->((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz)))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))) ((forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))->((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz))))->((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz)))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))) ((forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))->((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz))))->((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz)))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))) ((forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))->((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz))))->((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz)))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))) ((forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))->((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz))))->((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz)))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))) ((forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))->((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz))))->((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz)))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))) ((forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))->((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz))))->((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz)))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))) ((forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))->((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz))))->((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz)))))) b)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of b
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0))) ((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0))) ((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0))) ((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz)))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0))) ((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0))) ((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0))) ((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0))) ((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0))) ((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0))) ((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0))) ((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0))) ((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx) Xy)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx) Xy)))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x Xx) Xy))))
% Found (eta_expansion00 (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx) Xy)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx) Xy)))) b)
% Found ((eta_expansion0 Prop) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx) Xy)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx) Xy)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx) Xy)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx) Xy)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx) Xy)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx) Xy)))) b)
% Found (((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx) Xy)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx) Xy)))) b)
% Found x1:=(x (fun (x3:a) (x20:a)=> ((forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((Xp1 x3) x20))))):(((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))) ((forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))->((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz))))->((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))))))->((forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((Xp1 Xx) Xy))))
% Instantiate: b:=(((and (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((forall (Xx00:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy00))))->((Xp1 Xx00) Xy00)))->((forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))->((Xp1 Xx0) Xy0)))) ((forall (Xx0:a) (Xy00:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy00))))->((Xp1 Xx0) Xy00)))->((forall (Xx0:a) (Xy00:a) (Xz0:a), (((and ((Xp1 Xx0) Xy00)) ((Xp1 Xy00) Xz0))->((Xp1 Xx0) Xz0)))->((Xp1 Xy0) Xz))))->((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))))))->((forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((Xp1 Xx) Xy)))):Prop
% Found x1 as proof of b
% Found eta_expansion0000:=(eta_expansion000 (ex (a->(a->Prop)))):(((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and (S x)) ((x Xx0) Xy0)))))
% Found (eta_expansion000 (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found ((eta_expansion00 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found (((eta_expansion0 Prop) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found ((((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found ((((eta_expansion (a->(a->Prop))) Prop) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found eq_sym:=(fun (T:Type) (a:T) (b:T) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq T) x) a))) ((eq_ref T) a))):(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a)))
% Instantiate: b:=(forall (T:Type) (a:T) (b:T), ((((eq T) a) b)->(((eq T) b) a))):Prop
% Found eq_sym as proof of b
% Found x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Instantiate: x3:=Xp1:(a->(a->Prop))
% Found x2 as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((x3 Xx0) Xy0)) ((x3 Xy0) Xz))->((x3 Xx0) Xz)))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x3) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x3 Xx) Xy)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x3) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x3 Xx) Xy)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x3) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x3 Xx) Xy)))
% Found (fun (x3:(a->(a->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x3) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x3 Xx) Xy)))
% Found (fun (x3:(a->(a->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x Xx) Xy))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x3) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x3 Xx) Xy)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x3) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x3 Xx) Xy)))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x3) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x3 Xx) Xy)))
% Found (fun (x3:(a->(a->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x3) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x3 Xx) Xy)))
% Found (fun (x3:(a->(a->Prop)))=> ((eq_ref Prop) (f x3))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x Xx) Xy))))
% Found eq_ref00:=(eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xy0) Xz)))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xz))))))):(((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xy0) Xz)))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xz))))))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xy0) Xz)))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xz)))))))
% Found (eq_ref0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xy0) Xz)))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xz))))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xy0) Xz)))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xy0) Xz)))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xz))))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xy0) Xz)))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xy0) Xz)))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xz))))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xy0) Xz)))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xy0) Xz)))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xz))))))) as proof of (((eq Prop) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz))->((Xp100 Xx2) Xz))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xy0))))) ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xy0) Xz)))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp100:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp100 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp100 Xx2) Xy2)) ((Xp100 Xy2) Xz0))->((Xp100 Xx2) Xz0))))->((Xp100 Xx1) Xy1))))))))) ((R Xx0) Xz))))))) b)
% Found x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x2 as proof of b
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex (a->(a->Prop)))):(((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((ex (a->(a->Prop))) (fun (x:(a->(a->Prop)))=> ((and (S x)) ((x Xx0) Xy0)))))
% Found (eta_expansion_dep000 (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found ((eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found (((eta_expansion_dep0 (fun (x5:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found ((((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found eq_ref000:=(eq_ref00 (ex (a->(a->Prop)))):(((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))))
% Found (eq_ref00 (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found ((eq_ref0 (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found (((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found (((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))) (ex (a->(a->Prop)))) as proof of (P (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Found x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x3 as proof of b
% Found x3:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: x6:=(fun (x8:a) (x70:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x8) x70))))):(a->(a->Prop))
% Found x3 as proof of ((x6 Xx0) Xy0)
% Found x40000:=(x4000 x2):((Xp1 Xx0) Xz)
% Found (x4000 x2) as proof of ((Xp1 Xx0) Xz)
% Found ((x400 Xy0) x2) as proof of ((Xp1 Xx0) Xz)
% Found (((fun (Xy00:a)=> ((x40 Xy00) Xz)) Xy0) x2) as proof of ((Xp1 Xx0) Xz)
% Found (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2) as proof of ((Xp1 Xx0) Xz)
% Found (fun (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)) as proof of ((Xp1 Xx0) Xz)
% Found (fun (x3:(forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz0))->((Xp10 Xx2) Xz0))))->((Xp10 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)) as proof of ((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))
% Found (fun (x3:(forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz0))->((Xp10 Xx2) Xz0))))->((Xp10 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)) as proof of ((forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz0))->((Xp10 Xx2) Xz0))))->((Xp10 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz)))
% Found (and_rect00 (fun (x3:(forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz0))->((Xp10 Xx2) Xz0))))->((Xp10 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2))) as proof of ((Xp1 Xx0) Xz)
% Found ((and_rect0 ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz0))->((Xp10 Xx2) Xz0))))->((Xp10 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2))) as proof of ((Xp1 Xx0) Xz)
% Found (((fun (P:Type) (x3:((forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x3) x0)) ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz0))->((Xp10 Xx2) Xz0))))->((Xp10 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2))) as proof of ((Xp1 Xx0) Xz)
% Found (fun (x2:((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz)))=> (((fun (P:Type) (x3:((forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x3) x0)) ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz0))->((Xp10 Xx2) Xz0))))->((Xp10 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)))) as proof of ((Xp1 Xx0) Xz)
% Found (fun (Xz:a) (x2:((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz)))=> (((fun (P:Type) (x3:((forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x3) x0)) ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz0))->((Xp10 Xx2) Xz0))))->((Xp10 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)))) as proof of (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))
% Found (fun (Xy0:a) (Xz:a) (x2:((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz)))=> (((fun (P:Type) (x3:((forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x3) x0)) ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz0))->((Xp10 Xx2) Xz0))))->((Xp10 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)))) as proof of (forall (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found (fun (Xx0:a) (Xy0:a) (Xz:a) (x2:((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz)))=> (((fun (P:Type) (x3:((forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x3) x0)) ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz0))->((Xp10 Xx2) Xz0))))->((Xp10 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)))) as proof of (forall (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found (fun (Xx0:a) (Xy0:a) (Xz:a) (x2:((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz)))=> (((fun (P:Type) (x3:((forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x3) x0)) ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xx00:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz0:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz0))->((Xp10 Xx2) Xz0))))->((Xp10 Xx1) Xy1))))))))) ((R Xx00) Xy0))))->((Xp1 Xx00) Xy0)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)))) as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found x3:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: x6:=(fun (x8:a) (x70:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x8) x70))))):(a->(a->Prop))
% Found x3 as proof of ((x6 Xx0) Xy0)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found x5:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: x3:=(fun (x7:a) (x60:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x7) x60))))):(a->(a->Prop))
% Found (fun (x5:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))))=> x5) as proof of ((x3 Xx0) Xy0)
% Found (fun (Xy0:a) (x5:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))))=> x5) as proof of (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((x3 Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a) (x5:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))))=> x5) as proof of (forall (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((x3 Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x5:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0)))))=> x5) as proof of (forall (Xx0:a) (Xy0:a), (((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))->((x3 Xx0) Xy0)))
% Found x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x3 as proof of b
% Found x3:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: x4:=(fun (x8:a) (x70:a)=> ((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R x8) x70))))):(a->(a->Prop))
% Found x3 as proof of ((x4 Xx0) Xy0)
% Found eq_sym0:=(eq_sym Prop):(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a)))
% Instantiate: b:=(forall (a:Prop) (b:Prop), ((((eq Prop) a) b)->(((eq Prop) b) a))):Prop
% Found eq_sym0 as proof of b
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found x3:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: b:=(fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))):((a->(a->Prop))->Prop)
% Found x3 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x Xx0) Xy0))))
% Found (eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found ((eta_expansion_dep0 (fun (x5:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x5:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found x2000:=(x200 b):(((and ((Xp1 Xx) b)) ((Xp1 b) b))->((Xp1 Xx) b))
% Found (x200 b) as proof of (((and (P b)) (P b))->(P b))
% Found ((x20 b) b) as proof of (((and (P b)) (P b))->(P b))
% Found (((x2 Xx) b) b) as proof of (((and (P b)) (P b))->(P b))
% Found (((x2 Xx) b) b) as proof of (((and (P b)) (P b))->(P b))
% Found (fun (Xz:a)=> (((x2 Xx) b) b)) as proof of (((and (P b)) (P b))->(P b))
% Found (fun (Xy0:a) (Xz:a)=> (((x2 Xx) b) b)) as proof of (a->(((and (P b)) (P b))->(P b)))
% Found (fun (Xx0:a) (Xy0:a) (Xz:a)=> (((x2 Xx) b) b)) as proof of (a->(a->(((and (P b)) (P b))->(P b))))
% Found (fun (Xx0:a) (Xy0:a) (Xz:a)=> (((x2 Xx) b) b)) as proof of (a->(a->(a->(((and (P b)) (P b))->(P b)))))
% Found x4:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: b:=(fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x Xx0) Xy0))))
% Found (eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found ((eta_expansion_dep0 (fun (x6:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x6:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x6:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x6:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found eq_ref00:=(eq_ref0 Xy0):(((eq a) Xy0) Xy0)
% Found (eq_ref0 Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found ((eq_ref a) Xy0) as proof of (((eq a) Xy0) b)
% Found x4:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: b:=(fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) (fun (x:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x Xx0) Xy0))))
% Found (eta_expansion_dep00 (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found ((eta_expansion_dep0 (fun (x6:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x6:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x6:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found (((eta_expansion_dep (a->(a->Prop))) (fun (x6:(a->(a->Prop)))=> Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found x3:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: f:=(fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))):((a->(a->Prop))->Prop)
% Found x3 as proof of (P f)
% Found x3:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: f:=(fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))):((a->(a->Prop))->Prop)
% Found x3 as proof of (P f)
% Found x4:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: f:=(fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P f)
% Found x4:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: f:=(fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P f)
% Found x4:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: b:=(fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))
% Found (eq_ref0 (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x4) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x4 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x4) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x4 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x4) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x4 Xx0) Xy0)))
% Found (fun (x4:(a->(a->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x4) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x4 Xx0) Xy0)))
% Found (fun (x4:(a->(a->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x Xx0) Xy0))))
% Found eq_ref00:=(eq_ref0 (f x4)):(((eq Prop) (f x4)) (f x4))
% Found (eq_ref0 (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x4) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x4 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x4) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x4 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x4)) as proof of (((eq Prop) (f x4)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x4) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x4 Xx0) Xy0)))
% Found (fun (x4:(a->(a->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (((eq Prop) (f x4)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x4) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x4 Xx0) Xy0)))
% Found (fun (x4:(a->(a->Prop)))=> ((eq_ref Prop) (f x4))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x Xx0) Xy0))))
% Found x2000:=(x200 b):(((and ((Xp1 Xx) b)) ((Xp1 b) b))->((Xp1 Xx) b))
% Found (x200 b) as proof of (((and (P b)) (P b))->(P b))
% Found ((x20 b) b) as proof of (((and (P b)) (P b))->(P b))
% Found (((x2 Xx) b) b) as proof of (((and (P b)) (P b))->(P b))
% Found (((x2 Xx) b) b) as proof of (((and (P b)) (P b))->(P b))
% Found (fun (Xz:a)=> (((x2 Xx) b) b)) as proof of (((and (P b)) (P b))->(P b))
% Found (fun (Xy0:a) (Xz:a)=> (((x2 Xx) b) b)) as proof of (a->(((and (P b)) (P b))->(P b)))
% Found (fun (Xx0:a) (Xy0:a) (Xz:a)=> (((x2 Xx) b) b)) as proof of (a->(a->(((and (P b)) (P b))->(P b))))
% Found (fun (Xx0:a) (Xy0:a) (Xz:a)=> (((x2 Xx) b) b)) as proof of (a->(a->(a->(((and (P b)) (P b))->(P b)))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x5) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x5) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x5) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x5) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x Xx0) Xy0))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x5) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x5) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x5 Xx0) Xy0)))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x5) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x5) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x5 Xx0) Xy0)))
% Found (fun (x5:(a->(a->Prop)))=> ((eq_ref Prop) (f x5))) as proof of (forall (x:(a->(a->Prop))), (((eq Prop) (f x)) ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) x) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((x Xx0) Xy0))))
% Found x4:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: f:=(fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P f)
% Found x4:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: f:=(fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P f)
% Found x4:((ex (a->(a->Prop))) (fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))))
% Instantiate: b:=(fun (R:(a->(a->Prop)))=> ((and (S R)) ((R Xx0) Xy0))):((a->(a->Prop))->Prop)
% Found x4 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))):(((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0))))
% Found (eq_ref0 (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) as proof of (((eq ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (fun (Xx1:a) (Xy1:a)=> (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx2:a) (Xy2:a), (((Q Xx2) Xy2)->((Xp10 Xx2) Xy2)))) (forall (Xx2:a) (Xy2:a) (Xz:a), (((and ((Xp10 Xx2) Xy2)) ((Xp10 Xy2) Xz))->((Xp10 Xx2) Xz))))->((Xp10 Xx1) Xy1))))))))) ((R Xx0) Xy0)))) b)
% Found ((eq_ref ((a->(a->Prop))->Prop)) (fun (R:(a->(a->Prop)))=> ((and ((ex (a->(a->Prop))) (fun (Q:(a->(a->Prop)))=> ((and (S Q)) (((eq (a->(a->Prop))) R) (f
% EOF
%------------------------------------------------------------------------------