%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV098^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n106.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 13:33:44 EDT 2014

% Result   : Timeout 300.08s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV098^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n106.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 08:03:21 CDT 2014
% % CPUTime  : 300.08 
% 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 0x2055c68>, <kernel.Type object at 0x2055ea8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (Xr:(a->(a->Prop))) (Xs:a) (Xt:a), (((and (not (((eq a) Xs) Xt))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt))))->((ex a) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))))) of role conjecture named cTC_INTERP_OLD_pme
% Conjecture to prove = (forall (Xr:(a->(a->Prop))) (Xs:a) (Xt:a), (((and (not (((eq a) Xs) Xt))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt))))->((ex a) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (Xr:(a->(a->Prop))) (Xs:a) (Xt:a), (((and (not (((eq a) Xs) Xt))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt))))->((ex a) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))))']
% Parameter a:Type.
% Trying to prove (forall (Xr:(a->(a->Prop))) (Xs:a) (Xt:a), (((and (not (((eq a) Xs) Xt))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt))))->((ex a) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))))
% Found x1:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt)))
% Instantiate: x2:=Xs:a
% Found x1 as proof of (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))
% Found x2:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt)))
% Instantiate: x0:=Xs:a
% Found x2 as proof of (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) (fun (x:a)=> ((and ((Xr Xs) x)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x) Xt))))))
% Found (eta_expansion_dep00 (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) b)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found x3:((Xr Xx) Xy)
% Instantiate: x2:=Xy:a
% Found (fun (x3:((Xr Xx) Xy))=> x3) as proof of ((Xr Xx) x2)
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xs) x)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x) Xt))))))
% Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))))
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))))
% Found (fun (x0:a)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xs) x)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x) Xt))))))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found x3:((Xr Xx) Xy)
% Instantiate: x0:=Xy:a
% Found (fun (x3:((Xr Xx) Xy))=> x3) as proof of ((Xr Xx) x0)
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found (eq_ref0 (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt)))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xs) x)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x) Xt))))))
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))))
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))))
% Found (fun (x2:a)=> ((eq_ref Prop) (f x2))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xs) x)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x) Xt))))))
% Found x3:((Xr Xx) Xy)
% Instantiate: x2:=Xy:a
% Found x3 as proof of ((Xr Xx) x2)
% Found x3:((Xr Xx) Xy)
% Instantiate: x0:=Xy:a
% Found x3 as proof of ((Xr Xx) x0)
% Found x1:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt)))
% Instantiate: b:=(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt))):Prop
% Found x1 as proof of b
% Found x2:((Xr Xx) Xy)
% Instantiate: x3:=Xy:a
% Found x2 as proof of ((Xr Xx) x3)
% Found x4:((Xr Xx) x2)
% Found (fun (x5:((Xr Xy) x2))=> x4) as proof of ((Xr Xx) x2)
% Found (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4) as proof of (((Xr Xy) x2)->((Xr Xx) x2))
% Found (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4) as proof of (((Xr Xx) x2)->(((Xr Xy) x2)->((Xr Xx) x2)))
% Found (and_rect10 (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4)) as proof of ((Xr Xx) x2)
% Found ((and_rect1 ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4)) as proof of ((Xr Xx) x2)
% Found (((fun (P:Type) (x4:(((Xr Xx) x2)->(((Xr Xy) x2)->P)))=> (((((and_rect ((Xr Xx) x2)) ((Xr Xy) x2)) P) x4) x3)) ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4)) as proof of ((Xr Xx) x2)
% Found (fun (x3:((and ((Xr Xx) x2)) ((Xr Xy) x2)))=> (((fun (P:Type) (x4:(((Xr Xx) x2)->(((Xr Xy) x2)->P)))=> (((((and_rect ((Xr Xx) x2)) ((Xr Xy) x2)) P) x4) x3)) ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4))) as proof of ((Xr Xx) x2)
% Found (fun (Xz:a) (x3:((and ((Xr Xx) x2)) ((Xr Xy) x2)))=> (((fun (P:Type) (x4:(((Xr Xx) x2)->(((Xr Xy) x2)->P)))=> (((((and_rect ((Xr Xx) x2)) ((Xr Xy) x2)) P) x4) x3)) ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4))) as proof of (((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))
% Found (fun (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x2)) ((Xr Xy) x2)))=> (((fun (P:Type) (x4:(((Xr Xx) x2)->(((Xr Xy) x2)->P)))=> (((((and_rect ((Xr Xx) x2)) ((Xr Xy) x2)) P) x4) x3)) ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4))) as proof of (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x2)) ((Xr Xy) x2)))=> (((fun (P:Type) (x4:(((Xr Xx) x2)->(((Xr Xy) x2)->P)))=> (((((and_rect ((Xr Xx) x2)) ((Xr Xy) x2)) P) x4) x3)) ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4))) as proof of (forall (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x2)) ((Xr Xy) x2)))=> (((fun (P:Type) (x4:(((Xr Xx) x2)->(((Xr Xy) x2)->P)))=> (((((and_rect ((Xr Xx) x2)) ((Xr Xy) x2)) P) x4) x3)) ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4))) as proof of (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))
% Found x4:((Xr Xx) x2)
% Found (fun (x5:((Xr Xy) x2))=> x4) as proof of ((Xr Xx) x2)
% Found (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4) as proof of (((Xr Xy) x2)->((Xr Xx) x2))
% Found (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4) as proof of (((Xr Xx) x2)->(((Xr Xy) x2)->((Xr Xx) x2)))
% Found (and_rect10 (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4)) as proof of ((Xr Xx) x2)
% Found ((and_rect1 ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4)) as proof of ((Xr Xx) x2)
% Found (((fun (P:Type) (x4:(((Xr Xx) x2)->(((Xr Xy) x2)->P)))=> (((((and_rect ((Xr Xx) x2)) ((Xr Xy) x2)) P) x4) x3)) ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4)) as proof of ((Xr Xx) x2)
% Found (fun (x3:((and ((Xr Xx) x2)) ((Xr Xy) x2)))=> (((fun (P:Type) (x4:(((Xr Xx) x2)->(((Xr Xy) x2)->P)))=> (((((and_rect ((Xr Xx) x2)) ((Xr Xy) x2)) P) x4) x3)) ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4))) as proof of ((Xr Xx) x2)
% Found (fun (Xz:a) (x3:((and ((Xr Xx) x2)) ((Xr Xy) x2)))=> (((fun (P:Type) (x4:(((Xr Xx) x2)->(((Xr Xy) x2)->P)))=> (((((and_rect ((Xr Xx) x2)) ((Xr Xy) x2)) P) x4) x3)) ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4))) as proof of (((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))
% Found (fun (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x2)) ((Xr Xy) x2)))=> (((fun (P:Type) (x4:(((Xr Xx) x2)->(((Xr Xy) x2)->P)))=> (((((and_rect ((Xr Xx) x2)) ((Xr Xy) x2)) P) x4) x3)) ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4))) as proof of (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x2)) ((Xr Xy) x2)))=> (((fun (P:Type) (x4:(((Xr Xx) x2)->(((Xr Xy) x2)->P)))=> (((((and_rect ((Xr Xx) x2)) ((Xr Xy) x2)) P) x4) x3)) ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4))) as proof of (forall (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x2)) ((Xr Xy) x2)))=> (((fun (P:Type) (x4:(((Xr Xx) x2)->(((Xr Xy) x2)->P)))=> (((((and_rect ((Xr Xx) x2)) ((Xr Xy) x2)) P) x4) x3)) ((Xr Xx) x2)) (fun (x4:((Xr Xx) x2)) (x5:((Xr Xy) x2))=> x4))) as proof of (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))
% Found x4:((Xr Xx) Xx)
% Found (fun (x5:((Xr Xy) Xy))=> x4) as proof of ((Xr Xx) Xx)
% Found (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4) as proof of (((Xr Xy) Xy)->((Xr Xx) Xx))
% Found (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4) as proof of (((Xr Xx) Xx)->(((Xr Xy) Xy)->((Xr Xx) Xx)))
% Found (and_rect10 (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4)) as proof of ((Xr Xx) Xx)
% Found ((and_rect1 ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4)) as proof of ((Xr Xx) Xx)
% Found (((fun (P:Type) (x4:(((Xr Xx) Xx)->(((Xr Xy) Xy)->P)))=> (((((and_rect ((Xr Xx) Xx)) ((Xr Xy) Xy)) P) x4) x3)) ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4)) as proof of ((Xr Xx) Xx)
% Found (fun (x3:((and ((Xr Xx) Xx)) ((Xr Xy) Xy)))=> (((fun (P:Type) (x4:(((Xr Xx) Xx)->(((Xr Xy) Xy)->P)))=> (((((and_rect ((Xr Xx) Xx)) ((Xr Xy) Xy)) P) x4) x3)) ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4))) as proof of ((Xr Xx) Xx)
% Found (fun (Xz:a) (x3:((and ((Xr Xx) Xx)) ((Xr Xy) Xy)))=> (((fun (P:Type) (x4:(((Xr Xx) Xx)->(((Xr Xy) Xy)->P)))=> (((((and_rect ((Xr Xx) Xx)) ((Xr Xy) Xy)) P) x4) x3)) ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4))) as proof of (((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))
% Found (fun (Xy:a) (Xz:a) (x3:((and ((Xr Xx) Xx)) ((Xr Xy) Xy)))=> (((fun (P:Type) (x4:(((Xr Xx) Xx)->(((Xr Xy) Xy)->P)))=> (((((and_rect ((Xr Xx) Xx)) ((Xr Xy) Xy)) P) x4) x3)) ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4))) as proof of (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) Xx)) ((Xr Xy) Xy)))=> (((fun (P:Type) (x4:(((Xr Xx) Xx)->(((Xr Xy) Xy)->P)))=> (((((and_rect ((Xr Xx) Xx)) ((Xr Xy) Xy)) P) x4) x3)) ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4))) as proof of (forall (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) Xx)) ((Xr Xy) Xy)))=> (((fun (P:Type) (x4:(((Xr Xx) Xx)->(((Xr Xy) Xy)->P)))=> (((((and_rect ((Xr Xx) Xx)) ((Xr Xy) Xy)) P) x4) x3)) ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4))) as proof of (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))
% Found x4:((Xr Xx) x0)
% Found (fun (x5:((Xr Xy) x0))=> x4) as proof of ((Xr Xx) x0)
% Found (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4) as proof of (((Xr Xy) x0)->((Xr Xx) x0))
% Found (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4) as proof of (((Xr Xx) x0)->(((Xr Xy) x0)->((Xr Xx) x0)))
% Found (and_rect10 (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4)) as proof of ((Xr Xx) x0)
% Found ((and_rect1 ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4)) as proof of ((Xr Xx) x0)
% Found (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4)) as proof of ((Xr Xx) x0)
% Found (fun (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of ((Xr Xx) x0)
% Found (fun (Xz:a) (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of (((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))
% Found (fun (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of (forall (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref0 (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found x4:((Xr Xx) Xx)
% Found (fun (x5:((Xr Xy) Xy))=> x4) as proof of ((Xr Xx) Xx)
% Found (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4) as proof of (((Xr Xy) Xy)->((Xr Xx) Xx))
% Found (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4) as proof of (((Xr Xx) Xx)->(((Xr Xy) Xy)->((Xr Xx) Xx)))
% Found (and_rect10 (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4)) as proof of ((Xr Xx) Xx)
% Found ((and_rect1 ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4)) as proof of ((Xr Xx) Xx)
% Found (((fun (P:Type) (x4:(((Xr Xx) Xx)->(((Xr Xy) Xy)->P)))=> (((((and_rect ((Xr Xx) Xx)) ((Xr Xy) Xy)) P) x4) x3)) ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4)) as proof of ((Xr Xx) Xx)
% Found (fun (x3:((and ((Xr Xx) Xx)) ((Xr Xy) Xy)))=> (((fun (P:Type) (x4:(((Xr Xx) Xx)->(((Xr Xy) Xy)->P)))=> (((((and_rect ((Xr Xx) Xx)) ((Xr Xy) Xy)) P) x4) x3)) ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4))) as proof of ((Xr Xx) Xx)
% Found (fun (Xz:a) (x3:((and ((Xr Xx) Xx)) ((Xr Xy) Xy)))=> (((fun (P:Type) (x4:(((Xr Xx) Xx)->(((Xr Xy) Xy)->P)))=> (((((and_rect ((Xr Xx) Xx)) ((Xr Xy) Xy)) P) x4) x3)) ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4))) as proof of (((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))
% Found (fun (Xy:a) (Xz:a) (x3:((and ((Xr Xx) Xx)) ((Xr Xy) Xy)))=> (((fun (P:Type) (x4:(((Xr Xx) Xx)->(((Xr Xy) Xy)->P)))=> (((((and_rect ((Xr Xx) Xx)) ((Xr Xy) Xy)) P) x4) x3)) ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4))) as proof of (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) Xx)) ((Xr Xy) Xy)))=> (((fun (P:Type) (x4:(((Xr Xx) Xx)->(((Xr Xy) Xy)->P)))=> (((((and_rect ((Xr Xx) Xx)) ((Xr Xy) Xy)) P) x4) x3)) ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4))) as proof of (forall (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) Xx)) ((Xr Xy) Xy)))=> (((fun (P:Type) (x4:(((Xr Xx) Xx)->(((Xr Xy) Xy)->P)))=> (((((and_rect ((Xr Xx) Xx)) ((Xr Xy) Xy)) P) x4) x3)) ((Xr Xx) Xx)) (fun (x4:((Xr Xx) Xx)) (x5:((Xr Xy) Xy))=> x4))) as proof of (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))
% Found x4:((Xr Xx) x0)
% Found (fun (x5:((Xr Xy) x0))=> x4) as proof of ((Xr Xx) x0)
% Found (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4) as proof of (((Xr Xy) x0)->((Xr Xx) x0))
% Found (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4) as proof of (((Xr Xx) x0)->(((Xr Xy) x0)->((Xr Xx) x0)))
% Found (and_rect10 (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4)) as proof of ((Xr Xx) x0)
% Found ((and_rect1 ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4)) as proof of ((Xr Xx) x0)
% Found (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4)) as proof of ((Xr Xx) x0)
% Found (fun (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of ((Xr Xx) x0)
% Found (fun (Xz:a) (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of (((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))
% Found (fun (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of (forall (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->((ex a) (fun (x:a)=> ((and ((Xr Xy) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz)))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->((ex a) (fun (x:a)=> ((and ((Xr Xy) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz)))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eta_expansion_dep00 (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found x3:((Xr Xx) Xy)
% Instantiate: x0:=Xy:a
% Found (fun (x3:((Xr Xx) Xy))=> x3) as proof of ((Xr Xx) x0)
% Found eq_ref00:=(eq_ref0 x2):(((eq a) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq a) x2) Xy)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) Xy)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) Xy)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) Xy)
% Found (eq_sym0000 ((eq_ref a) x2)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found (eq_sym0000 ((eq_ref a) x2)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found ((fun (x3:(((eq a) x2) Xy))=> ((eq_sym000 x3) (Xr Xx))) ((eq_ref a) x2)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found ((fun (x3:(((eq a) x2) Xy))=> (((eq_sym00 Xy) x3) (Xr Xx))) ((eq_ref a) x2)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found ((fun (x3:(((eq a) x2) Xy))=> ((((eq_sym0 x2) Xy) x3) (Xr Xx))) ((eq_ref a) x2)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found ((fun (x3:(((eq a) x2) Xy))=> (((((eq_sym a) x2) Xy) x3) (Xr Xx))) ((eq_ref a) x2)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (P Xy)
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (P Xy)
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (P Xy)
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (P Xy)
% Found eq_ref00:=(eq_ref0 x2):(((eq a) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq a) x2) Xy)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) Xy)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) Xy)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) Xy)
% Found ((eq_sym0000 ((eq_ref a) x2)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found ((eq_sym0000 ((eq_ref a) x2)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found (((fun (x3:(((eq a) x2) Xy))=> ((eq_sym000 x3) (fun (x5:a)=> (((Xr Xx) Xy)->((Xr Xx) x5))))) ((eq_ref a) x2)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found (((fun (x3:(((eq a) x2) Xy))=> (((eq_sym00 Xy) x3) (fun (x5:a)=> (((Xr Xx) Xy)->((Xr Xx) x5))))) ((eq_ref a) x2)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found (((fun (x3:(((eq a) x2) Xy))=> ((((eq_sym0 x2) Xy) x3) (fun (x5:a)=> (((Xr Xx) Xy)->((Xr Xx) x5))))) ((eq_ref a) x2)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found (((fun (x3:(((eq a) x2) Xy))=> (((((eq_sym a) x2) Xy) x3) (fun (x5:a)=> (((Xr Xx) Xy)->((Xr Xx) x5))))) ((eq_ref a) x2)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found eq_ref00:=(eq_ref0 (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))):(((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))
% Found (eq_ref0 (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))) as proof of (((eq (a->Prop)) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))) b)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (P Xy)
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (P Xy)
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (P Xy)
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (P Xy)
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) Xy)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xy)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xy)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xy)
% Found (eq_sym0000 ((eq_ref a) x0)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (eq_sym0000 ((eq_ref a) x0)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found ((fun (x3:(((eq a) x0) Xy))=> ((eq_sym000 x3) (Xr Xx))) ((eq_ref a) x0)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found ((fun (x3:(((eq a) x0) Xy))=> (((eq_sym00 Xy) x3) (Xr Xx))) ((eq_ref a) x0)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found ((fun (x3:(((eq a) x0) Xy))=> ((((eq_sym0 x0) Xy) x3) (Xr Xx))) ((eq_ref a) x0)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found ((fun (x3:(((eq a) x0) Xy))=> (((((eq_sym a) x0) Xy) x3) (Xr Xx))) ((eq_ref a) x0)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of ((P b)->(P b))
% Found ((eq_ref0 b) P) as proof of ((P b)->(P b))
% Found (((eq_ref (a->Prop)) b) P) as proof of ((P b)->(P b))
% Found (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)) as proof of ((P b)->(P b))
% Found (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)) as proof of ((P b)->((P b)->(P b)))
% Found (and_rect10 (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P))) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P))) as proof of (P b)
% Found (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P))) as proof of (P b)
% Found (fun (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (P b)
% Found (fun (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (((and (P b)) (P b))->(P b))
% Found (fun (Xy:a) (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (a->(((and (P b)) (P b))->(P b)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (a->(a->(((and (P b)) (P b))->(P b))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (a->(a->(a->(((and (P b)) (P b))->(P b)))))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (P Xy)
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (P Xy)
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (P Xy)
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (P Xy)
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) Xy)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xy)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xy)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xy)
% Found ((eq_sym0000 ((eq_ref a) x0)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found ((eq_sym0000 ((eq_ref a) x0)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (((fun (x3:(((eq a) x0) Xy))=> ((eq_sym000 x3) (fun (x5:a)=> (((Xr Xx) Xy)->((Xr Xx) x5))))) ((eq_ref a) x0)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (((fun (x3:(((eq a) x0) Xy))=> (((eq_sym00 Xy) x3) (fun (x5:a)=> (((Xr Xx) Xy)->((Xr Xx) x5))))) ((eq_ref a) x0)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (((fun (x3:(((eq a) x0) Xy))=> ((((eq_sym0 x0) Xy) x3) (fun (x5:a)=> (((Xr Xx) Xy)->((Xr Xx) x5))))) ((eq_ref a) x0)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (((fun (x3:(((eq a) x0) Xy))=> (((((eq_sym a) x0) Xy) x3) (fun (x5:a)=> (((Xr Xx) Xy)->((Xr Xx) x5))))) ((eq_ref a) x0)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of ((P b)->(P b))
% Found ((eq_ref0 b) P) as proof of ((P b)->(P b))
% Found (((eq_ref (a->Prop)) b) P) as proof of ((P b)->(P b))
% Found (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)) as proof of ((P b)->(P b))
% Found (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)) as proof of ((P b)->((P b)->(P b)))
% Found (and_rect10 (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P))) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P))) as proof of (P b)
% Found (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P))) as proof of (P b)
% Found (fun (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (P b)
% Found (fun (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (((and (P b)) (P b))->(P b))
% Found (fun (Xy:a) (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (a->(((and (P b)) (P b))->(P b)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (a->(a->(((and (P b)) (P b))->(P b))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (a->(a->(a->(((and (P b)) (P b))->(P b)))))
% Found eq_ref00:=(eq_ref0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))):(((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (eq_ref0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (P Xy)
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (P Xy)
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (P Xy)
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (P Xy)
% Found eq_ref000:=(eq_ref00 P):((P f)->(P f))
% Found (eq_ref00 P) as proof of ((P f)->(P f))
% Found ((eq_ref0 f) P) as proof of ((P f)->(P f))
% Found (((eq_ref (a->Prop)) f) P) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)) as proof of ((P f)->((P f)->(P f)))
% Found (and_rect10 (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P))) as proof of (P f)
% Found (fun (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (P f)
% Found (fun (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (((and (P f)) (P f))->(P f))
% Found (fun (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (a->(((and (P f)) (P f))->(P f)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (a->(a->(((and (P f)) (P f))->(P f))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (a->(a->(a->(((and (P f)) (P f))->(P f)))))
% Found eq_ref000:=(eq_ref00 P):((P f)->(P f))
% Found (eq_ref00 P) as proof of ((P f)->(P f))
% Found ((eq_ref0 f) P) as proof of ((P f)->(P f))
% Found (((eq_ref (a->Prop)) f) P) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)) as proof of ((P f)->((P f)->(P f)))
% Found (and_rect10 (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P))) as proof of (P f)
% Found (fun (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (P f)
% Found (fun (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (((and (P f)) (P f))->(P f))
% Found (fun (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (a->(((and (P f)) (P f))->(P f)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (a->(a->(((and (P f)) (P f))->(P f))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (a->(a->(a->(((and (P f)) (P f))->(P f)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P f)->(P (fun (x:a)=> (f x))))
% Found (eta_expansion_dep000 P) as proof of ((P f)->(P f))
% Found ((eta_expansion_dep00 f) P) as proof of ((P f)->(P f))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) f) P) as proof of ((P f)->(P f))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)) as proof of ((P f)->((P f)->(P f)))
% Found (and_rect10 (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P))) as proof of (P f)
% Found (fun (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (P f)
% Found (fun (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (((and (P f)) (P f))->(P f))
% Found (fun (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (a->(((and (P f)) (P f))->(P f)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (a->(a->(((and (P f)) (P f))->(P f))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (a->(a->(a->(((and (P f)) (P f))->(P f)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P f)->(P (fun (x:a)=> (f x))))
% Found (eta_expansion_dep000 P) as proof of ((P f)->(P f))
% Found ((eta_expansion_dep00 f) P) as proof of ((P f)->(P f))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) f) P) as proof of ((P f)->(P f))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)) as proof of ((P f)->((P f)->(P f)))
% Found (and_rect10 (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P))) as proof of (P f)
% Found (fun (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (P f)
% Found (fun (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (((and (P f)) (P f))->(P f))
% Found (fun (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (a->(((and (P f)) (P f))->(P f)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (a->(a->(((and (P f)) (P f))->(P f))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (a->(a->(a->(((and (P f)) (P f))->(P f)))))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of ((P b)->(P b))
% Found ((eq_ref0 b) P) as proof of ((P b)->(P b))
% Found (((eq_ref (a->Prop)) b) P) as proof of ((P b)->(P b))
% Found (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)) as proof of ((P b)->(P b))
% Found (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)) as proof of ((P b)->((P b)->(P b)))
% Found (and_rect10 (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P))) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P))) as proof of (P b)
% Found (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P))) as proof of (P b)
% Found (fun (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (P b)
% Found (fun (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (((and (P b)) (P b))->(P b))
% Found (fun (Xy:a) (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (a->(((and (P b)) (P b))->(P b)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (a->(a->(((and (P b)) (P b))->(P b))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (a->(a->(a->(((and (P b)) (P b))->(P b)))))
% Found eq_ref000:=(eq_ref00 P):((P b)->(P b))
% Found (eq_ref00 P) as proof of ((P b)->(P b))
% Found ((eq_ref0 b) P) as proof of ((P b)->(P b))
% Found (((eq_ref (a->Prop)) b) P) as proof of ((P b)->(P b))
% Found (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)) as proof of ((P b)->(P b))
% Found (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)) as proof of ((P b)->((P b)->(P b)))
% Found (and_rect10 (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P))) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P))) as proof of (P b)
% Found (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P))) as proof of (P b)
% Found (fun (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (P b)
% Found (fun (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (((and (P b)) (P b))->(P b))
% Found (fun (Xy:a) (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (a->(((and (P b)) (P b))->(P b)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (a->(a->(((and (P b)) (P b))->(P b))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P b)) (P b)))=> (((fun (P0:Type) (x3:((P b)->((P b)->P0)))=> (((((and_rect (P b)) (P b)) P0) x3) x2)) (P b)) (fun (x3:(P b))=> (((eq_ref (a->Prop)) b) P)))) as proof of (a->(a->(a->(((and (P b)) (P b))->(P b)))))
% Found x3:((Xr Xx) Xy)
% Instantiate: x2:=Xx:a
% Found x3 as proof of ((Xr x2) Xy)
% Found (x400 x3) as proof of ((Xp0 x2) Xy)
% Found ((x40 Xy) x3) as proof of ((Xp0 x2) Xy)
% Found (((x4 x2) Xy) x3) as proof of ((Xp0 x2) Xy)
% Found (fun (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x2) Xy) x3)) as proof of ((Xp0 x2) Xy)
% Found (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x2) Xy) x3)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->((Xp0 x2) Xy))
% Found (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x2) Xy) x3)) as proof of ((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->((Xp0 x2) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x2) Xy) x3))) as proof of ((Xp0 x2) Xy)
% Found ((and_rect1 ((Xp0 x2) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x2) Xy) x3))) as proof of ((Xp0 x2) Xy)
% Found (((fun (P:Type) (x4:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))) P) x4) x00)) ((Xp0 x2) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x2) Xy) x3))) as proof of ((Xp0 x2) Xy)
% Found (fun (x00:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))))=> (((fun (P:Type) (x4:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))) P) x4) x00)) ((Xp0 x2) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x2) Xy) x3)))) as proof of ((Xp0 x2) Xy)
% Found (fun (Xp0:(a->(a->Prop))) (x00:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))))=> (((fun (P:Type) (x4:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))) P) x4) x00)) ((Xp0 x2) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x2) Xy) x3)))) as proof of (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))
% Found (fun (Xp0:(a->(a->Prop))) (x00:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))))=> (((fun (P:Type) (x4:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))) P) x4) x00)) ((Xp0 x2) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x2) Xy) x3)))) as proof of (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) 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 ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x3) Xy)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x3) Xy)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x3) Xy)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x3) Xy)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xx) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x) 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 ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x3) Xy)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x3) Xy)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x3) Xy)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x3) Xy)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xx) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x) Xy))))))
% Found eq_ref000:=(eq_ref00 P):((P f)->(P f))
% Found (eq_ref00 P) as proof of ((P f)->(P f))
% Found ((eq_ref0 f) P) as proof of ((P f)->(P f))
% Found (((eq_ref (a->Prop)) f) P) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)) as proof of ((P f)->((P f)->(P f)))
% Found (and_rect10 (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P))) as proof of (P f)
% Found (fun (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (P f)
% Found (fun (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (((and (P f)) (P f))->(P f))
% Found (fun (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (a->(((and (P f)) (P f))->(P f)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (a->(a->(((and (P f)) (P f))->(P f))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (a->(a->(a->(((and (P f)) (P f))->(P f)))))
% Found eq_ref000:=(eq_ref00 P):((P f)->(P f))
% Found (eq_ref00 P) as proof of ((P f)->(P f))
% Found ((eq_ref0 f) P) as proof of ((P f)->(P f))
% Found (((eq_ref (a->Prop)) f) P) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)) as proof of ((P f)->((P f)->(P f)))
% Found (and_rect10 (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P))) as proof of (P f)
% Found (fun (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (P f)
% Found (fun (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (((and (P f)) (P f))->(P f))
% Found (fun (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (a->(((and (P f)) (P f))->(P f)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (a->(a->(((and (P f)) (P f))->(P f))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> (((eq_ref (a->Prop)) f) P)))) as proof of (a->(a->(a->(((and (P f)) (P f))->(P f)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P f)->(P (fun (x:a)=> (f x))))
% Found (eta_expansion_dep000 P) as proof of ((P f)->(P f))
% Found ((eta_expansion_dep00 f) P) as proof of ((P f)->(P f))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) f) P) as proof of ((P f)->(P f))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)) as proof of ((P f)->((P f)->(P f)))
% Found (and_rect10 (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P))) as proof of (P f)
% Found (fun (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (P f)
% Found (fun (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (((and (P f)) (P f))->(P f))
% Found (fun (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (a->(((and (P f)) (P f))->(P f)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (a->(a->(((and (P f)) (P f))->(P f))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (a->(a->(a->(((and (P f)) (P f))->(P f)))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P):((P f)->(P (fun (x:a)=> (f x))))
% Found (eta_expansion_dep000 P) as proof of ((P f)->(P f))
% Found ((eta_expansion_dep00 f) P) as proof of ((P f)->(P f))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) f) P) as proof of ((P f)->(P f))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)) as proof of ((P f)->(P f))
% Found (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)) as proof of ((P f)->((P f)->(P f)))
% Found (and_rect10 (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P))) as proof of (P f)
% Found (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P))) as proof of (P f)
% Found (fun (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (P f)
% Found (fun (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (((and (P f)) (P f))->(P f))
% Found (fun (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (a->(((and (P f)) (P f))->(P f)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (a->(a->(((and (P f)) (P f))->(P f))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and (P f)) (P f)))=> (((fun (P0:Type) (x3:((P f)->((P f)->P0)))=> (((((and_rect (P f)) (P f)) P0) x3) x2)) (P f)) (fun (x3:(P f))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) f) P)))) as proof of (a->(a->(a->(((and (P f)) (P f))->(P f)))))
% Found x3:((Xr Xx) Xy)
% Instantiate: x0:=Xx:a
% Found x3 as proof of ((Xr x0) Xy)
% Found (x400 x3) as proof of ((Xp0 x0) Xy)
% Found ((x40 Xy) x3) as proof of ((Xp0 x0) Xy)
% Found (((x4 x0) Xy) x3) as proof of ((Xp0 x0) Xy)
% Found (fun (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x0) Xy) x3)) as proof of ((Xp0 x0) Xy)
% Found (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x0) Xy) x3)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->((Xp0 x0) Xy))
% Found (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x0) Xy) x3)) as proof of ((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->((Xp0 x0) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x0) Xy) x3))) as proof of ((Xp0 x0) Xy)
% Found ((and_rect1 ((Xp0 x0) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x0) Xy) x3))) as proof of ((Xp0 x0) Xy)
% Found (((fun (P:Type) (x4:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))) P) x4) x00)) ((Xp0 x0) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x0) Xy) x3))) as proof of ((Xp0 x0) Xy)
% Found (fun (x00:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))))=> (((fun (P:Type) (x4:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))) P) x4) x00)) ((Xp0 x0) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x0) Xy) x3)))) as proof of ((Xp0 x0) Xy)
% Found (fun (Xp0:(a->(a->Prop))) (x00:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))))=> (((fun (P:Type) (x4:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))) P) x4) x00)) ((Xp0 x0) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x0) Xy) x3)))) as proof of (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))
% Found (fun (Xp0:(a->(a->Prop))) (x00:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))))=> (((fun (P:Type) (x4:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))) P) x4) x00)) ((Xp0 x0) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x0) Xy) x3)))) as proof of (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) 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 ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x3) Xz)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x3) Xz)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x3) Xz)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x3) Xz)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xx) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz))))))
% 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 ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x3) Xz)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x3) Xz)))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((and ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x3) Xz)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((and ((Xr Xx) x3)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x3) Xz)))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xx) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz))))))
% Found x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))
% Found x7 as proof of (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))
% Found x6:((Xr Xx) x2)
% Found x6 as proof of ((Xr Xx) x2)
% Found x2:((Xr Xx) Xy)
% Instantiate: x3:=Xx:a
% Found x2 as proof of ((Xr x3) Xy)
% Found (x410 x2) as proof of ((Xp0 x3) Xy)
% Found ((x41 Xy) x2) as proof of ((Xp0 x3) Xy)
% Found (((x4 x3) Xy) x2) as proof of ((Xp0 x3) Xy)
% Found (fun (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x3) Xy) x2)) as proof of ((Xp0 x3) Xy)
% Found (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x3) Xy) x2)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->((Xp0 x3) Xy))
% Found (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x3) Xy) x2)) as proof of ((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->((Xp0 x3) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x3) Xy) x2))) as proof of ((Xp0 x3) Xy)
% Found ((and_rect1 ((Xp0 x3) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x3) Xy) x2))) as proof of ((Xp0 x3) Xy)
% Found (((fun (P:Type) (x4:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))) P) x4) x40)) ((Xp0 x3) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x3) Xy) x2))) as proof of ((Xp0 x3) Xy)
% Found (fun (x40:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))))=> (((fun (P:Type) (x4:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))) P) x4) x40)) ((Xp0 x3) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x3) Xy) x2)))) as proof of ((Xp0 x3) Xy)
% Found (fun (Xp0:(a->(a->Prop))) (x40:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))))=> (((fun (P:Type) (x4:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))) P) x4) x40)) ((Xp0 x3) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x3) Xy) x2)))) as proof of (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x3) Xy))
% Found (fun (Xp0:(a->(a->Prop))) (x40:((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))))=> (((fun (P:Type) (x4:((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0)))) P) x4) x40)) ((Xp0 x3) Xy)) (fun (x4:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (x5:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))=> (((x4 x3) Xy) x2)))) as proof of (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x3) Xy)))
% Found x10:=(x1 Xx):(forall (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))
% Found (x1 Xx) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((Xp x0) Xy)))
% Found (x1 Xx) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((Xp x0) Xy)))
% Found x30:=(x3 Xx):(forall (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))
% Found (x3 Xx) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((Xp x0) Xy)))
% Found (x3 Xx) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((Xp x0) Xy)))
% Found x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))
% Found x7 as proof of (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))
% Found eta_expansion_dep000:=(eta_expansion_dep00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion_dep00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion_dep0 (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x1:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion000:=(eta_expansion00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found ((eta_expansion0 Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found (((eta_expansion a) Prop) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found x6:((Xr Xx) x0)
% Found x6 as proof of ((Xr Xx) x0)
% Found x30:=(x3 Xx):(forall (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))
% Found (x3 Xx) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((Xp x0) Xy)))
% Found (x3 Xx) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((Xp x0) Xy)))
% Found x3:((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))))
% Instantiate: b:=(fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))):(a->Prop)
% Found x3 as proof of (P b)
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->((ex a) (fun (x:a)=> ((and ((Xr Xy) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz)))))))
% Found (eta_expansion000 (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P b))
% Found ((eta_expansion00 (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P b))
% Found (((eta_expansion0 Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P b))
% Found ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P b))
% Found ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P b))
% Found (fun (x3:((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))))=> ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a))) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P b))
% Found (fun (x3:((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))))=> ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a))) as proof of (((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))->(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P b)))
% Found (and_rect10 (fun (x3:((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))))=> ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)))) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x3:((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))))=> ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)))) as proof of (P b)
% Found (((fun (P0:Type) (x3:(((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))->(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->P0)))=> (((((and_rect ((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))) P0) x3) x2)) (P b)) (fun (x3:((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy)))))))=> ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)))) as proof of (P b)
% Found (((fun (P0:Type) (x3:(((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))->(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->P0)))=> (((((and_rect ((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))) P0) x3) x2)) (P b)) (fun (x3:((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy)))))))=> ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)))) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))):(((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (eq_ref0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))):(((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt))))
% Found (eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) b)
% Found x200:=(x20 Xy):(forall (Xz0:a), (((and ((Xp x0) Xy)) ((Xp Xy) Xz0))->((Xp x0) Xz0)))
% Found (x20 Xy) as proof of (forall (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found ((x2 x0) Xy) as proof of (forall (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found ((x2 x0) Xy) as proof of (forall (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found x100:=(x10 Xy):(((Xr Xx) Xy)->((Xp Xx) Xy))
% Found (x10 Xy) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found ((x1 Xx) Xy) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found ((x1 Xx) Xy) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found (fun (Xy:a)=> ((x1 Xx) Xy)) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found x400:=(x40 Xy):(forall (Xz0:a), (((and ((Xp x0) Xy)) ((Xp Xy) Xz0))->((Xp x0) Xz0)))
% Found (x40 Xy) as proof of (forall (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found ((x4 x0) Xy) as proof of (forall (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found ((x4 x0) Xy) as proof of (forall (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found x300:=(x30 Xy):(((Xr Xx) Xy)->((Xp Xx) Xy))
% Found (x30 Xy) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found ((x3 Xx) Xy) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found ((x3 Xx) Xy) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found (fun (Xy:a)=> ((x3 Xx) Xy)) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found x3:((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))))
% Instantiate: f:=(fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))):(a->Prop)
% Found x3 as proof of (P f)
% Found x3:((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))))
% Instantiate: f:=(fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))):(a->Prop)
% Found x3 as proof of (P f)
% Found x400:=(x40 Xy):(forall (Xz0:a), (((and ((Xp x0) Xy)) ((Xp Xy) Xz0))->((Xp x0) Xz0)))
% Found (x40 Xy) as proof of (forall (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found ((x4 x0) Xy) as proof of (forall (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found ((x4 x0) Xy) as proof of (forall (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found x300:=(x30 Xy):(((Xr Xx) Xy)->((Xp Xx) Xy))
% Found (x30 Xy) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found ((x3 Xx) Xy) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found ((x3 Xx) Xy) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found (fun (Xy:a)=> ((x3 Xx) Xy)) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found eta_expansion000:=(eta_expansion00 a0):(((eq (a->Prop)) a0) (fun (x:a)=> (a0 x)))
% Found (eta_expansion00 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eta_expansion0 Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion a) Prop) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (a->Prop)) b) (fun (x:a)=> (b x)))
% Found (eta_expansion_dep00 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz:a)=> ((and ((Xr Xs) Xz)) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp Xz) Xt))))))
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->((ex a) (fun (x:a)=> ((and ((Xr Xy) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz)))))))
% Found (eta_expansion000 (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f))
% Found ((eta_expansion00 (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f))
% Found (((eta_expansion0 Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f))
% Found ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f))
% Found ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f))
% Found (fun (x3:((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))))=> ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a))) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f))
% Found (fun (x3:((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))))=> ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a))) as proof of (((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))->(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f)))
% Found (and_rect10 (fun (x3:((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))))=> ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))))=> ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))->(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->P0)))=> (((((and_rect ((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))) P0) x3) x2)) (P f)) (fun (x3:((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy)))))))=> ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))->(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->P0)))=> (((((and_rect ((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))) P0) x3) x2)) (P f)) (fun (x3:((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy)))))))=> ((((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)))) as proof of (P f)
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->((ex a) (fun (x:a)=> ((and ((Xr Xy) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz)))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f))
% Found ((eta_expansion_dep00 (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f))
% Found (fun (x3:((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a))) as proof of (((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f))
% Found (fun (x3:((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a))) as proof of (((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))->(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P f)))
% Found (and_rect10 (fun (x3:((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy)))))))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))->(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->P0)))=> (((((and_rect ((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))) P0) x3) x2)) (P f)) (fun (x3:((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy)))))))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))->(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->P0)))=> (((((and_rect ((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))) P0) x3) x2)) (P f)) (fun (x3:((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy)))))))=> ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)))) as proof of (P f)
% Found eta_expansion000:=(eta_expansion00 (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))):(((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (fun (x:a)=> ((and ((Xr Xy) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz))))))
% Found (eta_expansion00 (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found x4:((Xr Xx) x0)
% Found (fun (x5:((Xr Xy) x0))=> x4) as proof of ((Xr Xx) x0)
% Found (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4) as proof of (((Xr Xy) x0)->((Xr Xx) x0))
% Found (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4) as proof of (((Xr Xx) x0)->(((Xr Xy) x0)->((Xr Xx) x0)))
% Found (and_rect10 (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4)) as proof of ((Xr Xx) x0)
% Found ((and_rect1 ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4)) as proof of ((Xr Xx) x0)
% Found (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4)) as proof of ((Xr Xx) x0)
% Found (fun (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of ((Xr Xx) x0)
% Found (fun (Xz:a) (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of (((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))
% Found (fun (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of (forall (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xr Xx) x0)) ((Xr Xy) x0)))=> (((fun (P:Type) (x4:(((Xr Xx) x0)->(((Xr Xy) x0)->P)))=> (((((and_rect ((Xr Xx) x0)) ((Xr Xy) x0)) P) x4) x3)) ((Xr Xx) x0)) (fun (x4:((Xr Xx) x0)) (x5:((Xr Xy) x0))=> x4))) as proof of (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))
% Found iff_refl:=(fun (A:Prop)=> ((((conj (A->A)) (A->A)) (fun (H:A)=> H)) (fun (H:A)=> H))):(forall (P:Prop), ((iff P) P))
% Instantiate: b:=(forall (P:Prop), ((iff P) P)):Prop
% Found iff_refl as proof of b
% Found x6:((Xr Xx) x2)
% Found (fun (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> x6) as proof of ((Xr Xx) x2)
% Found (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> x6) as proof of ((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->((Xr Xx) x2))
% Found (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> x6) as proof of (((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->((Xr Xx) x2)))
% Found (and_rect20 (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> x6)) as proof of ((Xr Xx) x2)
% Found ((and_rect2 ((Xr Xx) x2)) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> x6)) as proof of ((Xr Xx) x2)
% Found (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((Xr Xx) x2)) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> x6)) as proof of ((Xr Xx) x2)
% Found (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((Xr Xx) x2)) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> x6)) as proof of ((Xr Xx) x2)
% 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 ((Xr Xx) x5)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x5) Xz)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr Xx) x5)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x5) Xz)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr Xx) x5)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x5) Xz)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((Xr Xx) x5)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x5) Xz)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xx) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz))))))
% 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 ((Xr Xx) x5)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x5) Xz)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr Xx) x5)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x5) Xz)))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((and ((Xr Xx) x5)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x5) Xz)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((and ((Xr Xx) x5)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x5) Xz)))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xx) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref0 (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (ex a)) as proof of (P (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))):(((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (fun (x:a)=> ((and ((Xr Xy) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz))))))
% Found (eta_expansion00 (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eta_expansion0 Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion a) Prop) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found x2000:=(x200 Xz):(((and ((Xp x0) Xy)) ((Xp Xy) Xz))->((Xp x0) Xz))
% Found (x200 Xz) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found ((x20 Xy) Xz) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found (((x2 x0) Xy) Xz) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found (((x2 x0) Xy) Xz) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found (fun (Xz:a)=> (((x2 x0) Xy) Xz)) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found x4000:=(x400 Xz):(((and ((Xp x0) Xy)) ((Xp Xy) Xz))->((Xp x0) Xz))
% Found (x400 Xz) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found ((x40 Xy) Xz) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found (((x4 x0) Xy) Xz) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found (((x4 x0) Xy) Xz) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found (fun (Xz:a)=> (((x4 x0) Xy) Xz)) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found conj100:=(conj10 (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))):((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))
% Found (conj10 (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) as proof of ((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))
% Found ((fun (B:Prop)=> ((conj1 B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) as proof of ((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))
% Found ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) as proof of ((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))
% Found (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) as proof of ((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))
% Found (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) as proof of (((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))
% Found (and_rect30 (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))) as proof of ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))
% Found ((and_rect3 ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))) as proof of ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))
% Found (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))) as proof of ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))
% Found (fun (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))) as proof of ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))
% Found (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))) as proof of ((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))
% Found (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))) as proof of (((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))
% Found (and_rect20 (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))))) as proof of ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))
% Found ((and_rect2 ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))))) as proof of ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))
% Found (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))))) as proof of ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))
% Found (fun (x5:((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))))) as proof of ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))
% Found (fun (x4:((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) (x5:((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))))) as proof of (((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))
% Found (fun (x4:((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) (x5:((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))))) as proof of (((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))->(((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))
% Found (and_rect10 (fun (x4:((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) (x5:((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))))))) as proof of ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))
% Found ((and_rect1 ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x4:((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) (x5:((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))))))) as proof of ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))
% Found (((fun (P:Type) (x4:(((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))->(((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))->P)))=> (((((and_rect ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) P) x4) x3)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x4:((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) (x5:((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))))))) as proof of ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))
% Found (fun (x3:((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz))))))=> (((fun (P:Type) (x4:(((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))->(((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))->P)))=> (((((and_rect ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) P) x4) x3)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x4:((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) (x5:((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))))))) as proof of ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))
% Found (fun (Xz:a) (x3:((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz))))))=> (((fun (P:Type) (x4:(((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))->(((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))->P)))=> (((((and_rect ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) P) x4) x3)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x4:((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) (x5:((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))))))) as proof of (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))
% Found (fun (Xy:a) (Xz:a) (x3:((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz))))))=> (((fun (P:Type) (x4:(((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))->(((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))->P)))=> (((((and_rect ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) P) x4) x3)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x4:((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) (x5:((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))))))) as proof of (forall (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz))))))=> (((fun (P:Type) (x4:(((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))->(((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))->P)))=> (((((and_rect ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) P) x4) x3)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x4:((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) (x5:((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))))))) as proof of (forall (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz))))))=> (((fun (P:Type) (x4:(((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))->(((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))->P)))=> (((((and_rect ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) P) x4) x3)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x4:((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) (x5:((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))->P)))=> (((((and_rect ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) P) x6) x4)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x6:((Xr Xx) x2)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x2)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))->P)))=> (((((and_rect ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))) P) x8) x5)) ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))) (fun (x8:((Xr Xy) x2))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x2)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))))))) as proof of (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq Prop) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found ((eq_ref Prop) a0) as proof of (((eq Prop) a0) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((ex a) (fun (Xz:a)=> ((and ((Xr Xx) Xz)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 Xz) Xy))))))) ((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx) Xz00))))->((Xp0 Xz0) Xz)))))))->((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))))))
% Found eq_ref000:=(eq_ref00 (Xp x0)):(((Xp x0) Xz)->((Xp x0) Xz))
% Found (eq_ref00 (Xp x0)) as proof of (((Xp x0) Xz)->((Xp x0) Xz))
% Found ((eq_ref0 Xz) (Xp x0)) as proof of (((Xp x0) Xz)->((Xp x0) Xz))
% Found (((eq_ref a) Xz) (Xp x0)) as proof of (((Xp x0) Xz)->((Xp x0) Xz))
% Found (fun (x4:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))) as proof of (((Xp x0) Xz)->((Xp x0) Xz))
% Found (fun (x4:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))) as proof of (((Xp x0) Xy)->(((Xp x0) Xz)->((Xp x0) Xz)))
% Found (and_rect10 (fun (x4:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0)))) as proof of ((Xp x0) Xz)
% Found ((and_rect1 ((Xp x0) Xz)) (fun (x4:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0)))) as proof of ((Xp x0) Xz)
% Found (((fun (P:Type) (x4:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x4) x3)) ((Xp x0) Xz)) (fun (x4:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0)))) as proof of ((Xp x0) Xz)
% Found (fun (x3:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x4:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x4) x3)) ((Xp x0) Xz)) (fun (x4:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of ((Xp x0) Xz)
% Found (fun (Xz:a) (x3:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x4:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x4) x3)) ((Xp x0) Xz)) (fun (x4:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found (fun (Xy:a) (Xz:a) (x3:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x4:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x4) x3)) ((Xp x0) Xz)) (fun (x4:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of (forall (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x4:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x4) x3)) ((Xp x0) Xz)) (fun (x4:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of (forall (Xy:a) (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x4:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x4) x3)) ((Xp x0) Xz)) (fun (x4:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of (a->(forall (Xy:a) (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))))
% Found eq_ref00:=(eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))):(((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt))))
% Found (eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x2) Xt)))) b)
% Found x4000:=(x400 Xz):(((and ((Xp x0) Xy)) ((Xp Xy) Xz))->((Xp x0) Xz))
% Found (x400 Xz) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found ((x40 Xy) Xz) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found (((x4 x0) Xy) Xz) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found (((x4 x0) Xy) Xz) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found (fun (Xz:a)=> (((x4 x0) Xy) Xz)) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found x6:((Xr Xx) x0)
% Found (fun (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> x6) as proof of ((Xr Xx) x0)
% Found (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> x6) as proof of ((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->((Xr Xx) x0))
% Found (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> x6) as proof of (((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->((Xr Xx) x0)))
% Found (and_rect20 (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> x6)) as proof of ((Xr Xx) x0)
% Found ((and_rect2 ((Xr Xx) x0)) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> x6)) as proof of ((Xr Xx) x0)
% Found (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((Xr Xx) x0)) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> x6)) as proof of ((Xr Xx) x0)
% Found (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((Xr Xx) x0)) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> x6)) as proof of ((Xr Xx) x0)
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: a0:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of a0
% Found conj100:=(conj10 (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))):((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))
% Found (conj10 (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) as proof of ((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))
% Found ((fun (B:Prop)=> ((conj1 B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) as proof of ((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))
% Found ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) as proof of ((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))
% Found (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) as proof of ((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))
% Found (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) as proof of (((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))
% Found (and_rect30 (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))) as proof of ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))
% Found ((and_rect3 ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))) as proof of ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))
% Found (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))) as proof of ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))
% Found (fun (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))) as proof of ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))
% Found (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))) as proof of ((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))
% Found (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))) as proof of (((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))
% Found (and_rect20 (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))))) as proof of ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))
% Found ((and_rect2 ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))))) as proof of ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))
% Found (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))))) as proof of ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))
% Found (fun (x5:((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))))) as proof of ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))
% Found (fun (x4:((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) (x5:((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))))) as proof of (((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))
% Found (fun (x4:((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) (x5:((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))))) as proof of (((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))->(((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))
% Found (and_rect10 (fun (x4:((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) (x5:((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))))))) as proof of ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))
% Found ((and_rect1 ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x4:((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) (x5:((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))))))) as proof of ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))
% Found (((fun (P:Type) (x4:(((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))->(((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))->P)))=> (((((and_rect ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) P) x4) x3)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x4:((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) (x5:((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))))))) as proof of ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))
% Found (fun (x3:((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz))))))=> (((fun (P:Type) (x4:(((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))->(((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))->P)))=> (((((and_rect ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) P) x4) x3)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x4:((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) (x5:((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))))))) as proof of ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))
% Found (fun (Xz:a) (x3:((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz))))))=> (((fun (P:Type) (x4:(((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))->(((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))->P)))=> (((((and_rect ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) P) x4) x3)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x4:((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) (x5:((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))))))) as proof of (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))
% Found (fun (Xy:a) (Xz:a) (x3:((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz))))))=> (((fun (P:Type) (x4:(((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))->(((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))->P)))=> (((((and_rect ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) P) x4) x3)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x4:((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) (x5:((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))))))) as proof of (forall (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz))))))=> (((fun (P:Type) (x4:(((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))->(((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))->P)))=> (((((and_rect ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) P) x4) x3)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x4:((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) (x5:((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))))))) as proof of (forall (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz))))))=> (((fun (P:Type) (x4:(((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))->(((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))->P)))=> (((((and_rect ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) P) x4) x3)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x4:((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) (x5:((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))=> (((fun (P:Type) (x6:(((Xr Xx) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))->P)))=> (((((and_rect ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) P) x6) x4)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x6:((Xr Xx) x0)) (x7:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))=> (((fun (P:Type) (x8:(((Xr Xy) x0)->((forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))->P)))=> (((((and_rect ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))) P) x8) x5)) ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))) (fun (x8:((Xr Xy) x0))=> ((fun (B:Prop)=> (((conj ((Xr Xx) x0)) B) x6)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))))))) as proof of (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))
% Found eq_ref00:=(eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))):(((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt))))
% Found (eq_ref0 (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) b)
% Found ((eq_ref Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) as proof of (((eq Prop) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0))))->((Xp x0) Xt)))) b)
% Found x0:(not (((eq a) Xs) Xt))
% Instantiate: b:=((((eq a) Xs) Xt)->False):Prop
% Found x0 as proof of a0
% Found ex_intro00:=(ex_intro0 (Xr Xx)):(forall (x:a), (((Xr Xx) x)->((ex a) (Xr Xx))))
% Found (ex_intro0 (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) b)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) b)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) b)))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz)))))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))) ((and ((Xr Xy) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x2) Xz)))))->((and ((Xr Xx) x2)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xz))))))) b)
% Found ex_intro00:=(ex_intro0 (Xr Xx)):(forall (x:a), (((Xr Xx) x)->((ex a) (Xr Xx))))
% Found (ex_intro0 (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ex_intro00:=(ex_intro0 (Xr Xx)):(forall (x:a), (((Xr Xx) x)->((ex a) (Xr Xx))))
% Found (ex_intro0 (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ex_intro000:=(ex_intro00 Xy):(((Xr Xx) Xy)->((ex a) (Xr Xx)))
% Found (ex_intro00 Xy) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found ((ex_intro0 (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found (fun (Xy:a)=> (((ex_intro a) (Xr Xx)) Xy)) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found x3:((Xr Xx) Xy)
% Instantiate: x0:=Xx:a
% Found x3 as proof of ((Xr x0) Xy)
% Found (x400 x3) as proof of ((Xp x0) Xy)
% Found ((x40 Xy) x3) as proof of ((Xp x0) Xy)
% Found (((x4 x0) Xy) x3) as proof of ((Xp x0) Xy)
% Found (fun (x5:(forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx) Xz0))))=> (((x4 x0) Xy) x3)) as proof of ((Xp x0) Xy)
% Found (fun (x4:(forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp Xx) Xy0)))) (x5:(forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx) Xz0))))=> (((x4 x0) Xy) x3)) as proof of ((forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx) Xz0)))->((Xp x0) Xy))
% Found (fun (x4:(forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp Xx) Xy0)))) (x5:(forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx) Xz0))))=> (((x4 x0) Xy) x3)) as proof of ((forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp Xx) Xy0)))->((forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx) Xz0)))->((Xp x0) Xy)))
% Found (and_rect10 (fun (x4:(forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp Xx) Xy0)))) (x5:(forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx) Xz0))))=> (((x4 x0) Xy) x3))) as proof of ((Xp x0) Xy)
% Found ((and_rect1 ((Xp x0) Xy)) (fun (x4:(forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp Xx) Xy0)))) (x5:(forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx) Xz0))))=> (((x4 x0) Xy) x3))) as proof of ((Xp x0) Xy)
% Found (((fun (P:Type) (x4:((forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))->((forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0)))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0)))) P) x4) x00)) ((Xp x0) Xy)) (fun (x4:(forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp Xx) Xy0)))) (x5:(forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx) Xz0))))=> (((x4 x0) Xy) x3))) as proof of ((Xp x0) Xy)
% Found (fun (x3:((Xr Xx) Xy))=> (((fun (P:Type) (x4:((forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))->((forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0)))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0)))) P) x4) x00)) ((Xp x0) Xy)) (fun (x4:(forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp Xx) Xy0)))) (x5:(forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx) Xz0))))=> (((x4 x0) Xy) x3)))) as proof of ((Xp x0) Xy)
% Found (fun (Xy:a) (x3:((Xr Xx) Xy))=> (((fun (P:Type) (x4:((forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))->((forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0)))->P)))=> (((((and_rect (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz0:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz0))->((Xp Xx) Xz0)))) P) x4) x00)) ((Xp x0) Xy)) (fun (x4:(forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp Xx) Xy0)))) (x5:(forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx) Xz0))))=> (((x4 x0) Xy) x3)))) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found ex_intro00:=(ex_intro0 (Xr Xx)):(forall (x:a), (((Xr Xx) x)->((ex a) (Xr Xx))))
% Found (ex_intro0 (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) b)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) b)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) b)))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))):(((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz)))))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))) ((and ((Xr Xy) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx:a) (Xy0:a), (((Xr Xx) Xy0)->((Xp0 Xx) Xy0)))) (forall (Xx:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx) Xz0))))->((Xp0 x0) Xz)))))->((and ((Xr Xx) x0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz0:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xz))))))) b)
% Found ex_intro00:=(ex_intro0 (Xr Xx)):(forall (x:a), (((Xr Xx) x)->((ex a) (Xr Xx))))
% Found (ex_intro0 (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) b)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) b)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) b)))
% Found x1:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt)))
% Instantiate: x2:=Xs:a
% Found x1 as proof of b
% Found ex_intro000:=(ex_intro00 Xy):(((Xr Xx) Xy)->((ex a) (Xr Xx)))
% Found (ex_intro00 Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found ((ex_intro0 (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (fun (Xy:a)=> (((ex_intro a) (Xr Xx)) Xy)) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found ex_intro000:=(ex_intro00 Xy):(((Xr Xx) Xy)->((ex a) (Xr Xx)))
% Found (ex_intro00 Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found ((ex_intro0 (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (fun (Xy:a)=> (((ex_intro a) (Xr Xx)) Xy)) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found ex_intro00:=(ex_intro0 (Xr Xx)):(forall (x:a), (((Xr Xx) x)->((ex a) (Xr Xx))))
% Found (ex_intro0 (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ex_intro00:=(ex_intro0 (Xr Xx)):(forall (x:a), (((Xr Xx) x)->((ex a) (Xr Xx))))
% Found (ex_intro0 (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found x5:((Xr Xx) Xy)
% Instantiate: x0:=Xx:a
% Found (fun (x5:((Xr Xx) Xy))=> x5) as proof of ((Xr x0) Xy)
% Found (fun (Xy:a) (x5:((Xr Xx) Xy))=> x5) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found x5:((Xr Xx) Xy)
% Instantiate: x0:=Xx:a
% Found (fun (x5:((Xr Xx) Xy))=> x5) as proof of ((Xr x0) Xy)
% Found (fun (Xy:a) (x5:((Xr Xx) Xy))=> x5) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found ex_intro00:=(ex_intro0 (Xr Xx)):(forall (x:a), (((Xr Xx) x)->((ex a) (Xr Xx))))
% Found (ex_intro0 (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ex_intro00:=(ex_intro0 (Xr Xx)):(forall (x:a), (((Xr Xx) x)->((ex a) (Xr Xx))))
% Found (ex_intro0 (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found ((ex_intro a) (Xr Xx)) as proof of (forall (Xy:a), (((Xr Xx) Xy)->((ex a) f)))
% Found eq_ref00:=(eq_ref0 x2):(((eq a) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq a) x2) b)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) b)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) b)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) b)
% Found ex_intro000:=(ex_intro00 Xy):(((Xr Xx) Xy)->((ex a) (Xr Xx)))
% Found (ex_intro00 Xy) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found ((ex_intro0 (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found (fun (Xy:a)=> (((ex_intro a) (Xr Xx)) Xy)) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found (fun (Xy:a)=> (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found x2:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt)))
% Instantiate: x0:=Xs:a
% Found x2 as proof of b
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found (fun (Xy:a)=> (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found ex_intro000:=(ex_intro00 Xy):(((Xr Xx) Xy)->((ex a) (Xr Xx)))
% Found (ex_intro00 Xy) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found ((ex_intro0 (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found (fun (Xy:a)=> (((ex_intro a) (Xr Xx)) Xy)) as proof of (((Xr Xx) Xy)->((ex a) b))
% Found eq_ref00:=(eq_ref0 x2):(((eq a) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq a) x2) b)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) b)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) b)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) b)
% Found eq_ref00:=(eq_ref0 x2):(((eq a) x2) x2)
% Found (eq_ref0 x2) as proof of (((eq a) x2) b)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) b)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) b)
% Found ((eq_ref a) x2) as proof of (((eq a) x2) b)
% Found x5:((Xr Xx) Xy)
% Instantiate: x0:=Xx:a
% Found (fun (x5:((Xr Xx) Xy))=> x5) as proof of ((Xr x0) Xy)
% Found (fun (Xy:a) (x5:((Xr Xx) Xy))=> x5) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% 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_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found (fun (Xy:a)=> (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr x0) Xy))
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found ex_intro000:=(ex_intro00 Xy):(((Xr Xx) Xy)->((ex a) (Xr Xx)))
% Found (ex_intro00 Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found ((ex_intro0 (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (fun (Xy:a)=> (((ex_intro a) (Xr Xx)) Xy)) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found ex_intro000:=(ex_intro00 Xy):(((Xr Xx) Xy)->((ex a) (Xr Xx)))
% Found (ex_intro00 Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found ((ex_intro0 (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (fun (Xy:a)=> (((ex_intro a) (Xr Xx)) Xy)) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (P Xy)
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (P Xy)
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (P Xy)
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (P Xy)
% Found ex_intro000:=(ex_intro00 Xy):(((Xr Xx) Xy)->((ex a) (Xr Xx)))
% Found (ex_intro00 Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found ((ex_intro0 (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (fun (Xy:a)=> (((ex_intro a) (Xr Xx)) Xy)) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found ex_intro000:=(ex_intro00 Xy):(((Xr Xx) Xy)->((ex a) (Xr Xx)))
% Found (ex_intro00 Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found ((ex_intro0 (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (((ex_intro a) (Xr Xx)) Xy) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found (fun (Xy:a)=> (((ex_intro a) (Xr Xx)) Xy)) as proof of (((Xr Xx) Xy)->((ex a) f))
% Found or_ind:(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P))))
% Instantiate: b:=(forall (A:Prop) (B:Prop) (P:Prop), ((A->P)->((B->P)->(((or A) B)->P)))):Prop
% Found or_ind as proof of b
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) Xy)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xy)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xy)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xy)
% Found ((eq_sym0000 ((eq_ref a) x0)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found ((eq_sym0000 ((eq_ref a) x0)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (((fun (x3:(((eq a) x0) Xy))=> ((eq_sym000 x3) (fun (x5:a)=> (((Xr Xx) Xy)->((Xr Xx) x5))))) ((eq_ref a) x0)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (((fun (x3:(((eq a) x0) Xy))=> (((eq_sym00 Xy) x3) (fun (x5:a)=> (((Xr Xx) Xy)->((Xr Xx) x5))))) ((eq_ref a) x0)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (((fun (x3:(((eq a) x0) Xy))=> ((((eq_sym0 x0) Xy) x3) (fun (x5:a)=> (((Xr Xx) Xy)->((Xr Xx) x5))))) ((eq_ref a) x0)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (((fun (x3:(((eq a) x0) Xy))=> (((((eq_sym a) x0) Xy) x3) (fun (x5:a)=> (((Xr Xx) Xy)->((Xr Xx) x5))))) ((eq_ref a) x0)) (((eq_ref a) Xy) (Xr Xx))) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) Xy)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xy)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xy)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) Xy)
% Found (eq_sym0000 ((eq_ref a) x0)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found (eq_sym0000 ((eq_ref a) x0)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found ((fun (x3:(((eq a) x0) Xy))=> ((eq_sym000 x3) (Xr Xx))) ((eq_ref a) x0)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found ((fun (x3:(((eq a) x0) Xy))=> (((eq_sym00 Xy) x3) (Xr Xx))) ((eq_ref a) x0)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found ((fun (x3:(((eq a) x0) Xy))=> ((((eq_sym0 x0) Xy) x3) (Xr Xx))) ((eq_ref a) x0)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found ((fun (x3:(((eq a) x0) Xy))=> (((((eq_sym a) x0) Xy) x3) (Xr Xx))) ((eq_ref a) x0)) as proof of (((Xr Xx) Xy)->((Xr Xx) x0))
% Found x3:((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))))
% Instantiate: b:=(fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))):(a->Prop)
% Found x3 as proof of (P b)
% Found x3:((ex a) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))))
% Instantiate: b:=(fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xy))))):(a->Prop)
% Found x3 as proof of (P b)
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found x2:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt)))
% Instantiate: x0:=Xs:a
% Found x2 as proof of b
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))):(((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (fun (x:a)=> ((and ((Xr Xx) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x8:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x8:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) 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 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))):(((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (eq_ref0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))):(((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2)))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) b)
% Found x5:((Xr Xx) Xy)
% Instantiate: x0:=Xx:a
% Found x5 as proof of ((Xr x0) Xy)
% Found (x100 x5) as proof of ((Xp x0) Xy)
% Found ((x10 Xy) x5) as proof of ((Xp x0) Xy)
% Found (((x1 x0) Xy) x5) as proof of ((Xp x0) Xy)
% Found (fun (x5:((Xr Xx) Xy))=> (((x1 x0) Xy) x5)) as proof of ((Xp x0) Xy)
% Found (fun (Xy:a) (x5:((Xr Xx) Xy))=> (((x1 x0) Xy) x5)) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found x5:((Xr Xx) Xy)
% Instantiate: x0:=Xx:a
% Found x5 as proof of ((Xr x0) Xy)
% Found (x300 x5) as proof of ((Xp x0) Xy)
% Found ((x30 Xy) x5) as proof of ((Xp x0) Xy)
% Found (((x3 x0) Xy) x5) as proof of ((Xp x0) Xy)
% Found (fun (x5:((Xr Xx) Xy))=> (((x3 x0) Xy) x5)) as proof of ((Xp x0) Xy)
% Found (fun (Xy:a) (x5:((Xr Xx) Xy))=> (((x3 x0) Xy) x5)) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found eq_ref00:=(eq_ref0 x0):(((eq a) x0) x0)
% Found (eq_ref0 x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found ((eq_ref a) x0) as proof of (((eq a) x0) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))):(((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2)))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x2)) ((Xr Xy) x2))->((Xr Xx) x2))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))):(((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx)))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) b)
% Found x4:((Xr Xx) Xy)
% Instantiate: x0:=Xx:a
% Found x4 as proof of ((Xr x0) Xy)
% Found (x300 x4) as proof of ((Xp x0) Xy)
% Found ((x30 Xy) x4) as proof of ((Xp x0) Xy)
% Found (((x3 x0) Xy) x4) as proof of ((Xp x0) Xy)
% Found (fun (x40:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx0) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx0) Xz0))))=> (((x3 x0) Xy) x4)) as proof of ((Xp x0) Xy)
% Found (fun (x4:((Xr Xx) Xy)) (x40:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx0) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx0) Xz0))))=> (((x3 x0) Xy) x4)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx0) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx0) Xz0)))->((Xp x0) Xy))
% Found (fun (Xy:a) (x4:((Xr Xx) Xy)) (x40:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx0) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx0) Xz0))))=> (((x3 x0) Xy) x4)) as proof of (((Xr Xx) Xy)->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx0) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx0) Xz0)))->((Xp x0) Xy)))
% Found x5:((Xr Xx) Xy)
% Instantiate: x0:=Xx:a
% Found x5 as proof of ((Xr x0) Xy)
% Found (x300 x5) as proof of ((Xp x0) Xy)
% Found ((x30 Xy) x5) as proof of ((Xp x0) Xy)
% Found (((x3 x0) Xy) x5) as proof of ((Xp x0) Xy)
% Found (fun (x5:((Xr Xx) Xy))=> (((x3 x0) Xy) x5)) as proof of ((Xp x0) Xy)
% Found (fun (Xy:a) (x5:((Xr Xx) Xy))=> (((x3 x0) Xy) x5)) as proof of (((Xr Xx) Xy)->((Xp x0) Xy))
% Found x4:((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Instantiate: f:=(fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))):(a->Prop)
% Found x4 as proof of (P f)
% Found x4:((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Instantiate: f:=(fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))):(a->Prop)
% Found x4 as proof of (P f)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))):(((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0)))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) b)
% Found x4:((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Instantiate: f:=(fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))):(a->Prop)
% Found x4 as proof of (P f)
% Found x4:((ex a) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Instantiate: f:=(fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))):(a->Prop)
% Found x4 as proof of (P f)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found x3:((Xr Xx) Xy)
% Instantiate: x0:=Xx:a
% Found x3 as proof of ((Xr x0) Xy)
% Found (x3000 x3) as proof of ((Xp x0) Xy)
% Found ((x300 Xy) x3) as proof of ((Xp x0) Xy)
% Found (((x30 x0) Xy) x3) as proof of ((Xp x0) Xy)
% Found (fun (x400:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx0) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx0) Xz0))))=> (((x30 x0) Xy) x3)) as proof of ((Xp x0) Xy)
% Found (fun (x30:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x400:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx0) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx0) Xz0))))=> (((x30 x0) Xy) x3)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx0) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx0) Xz0)))->((Xp x0) Xy))
% Found (fun (x3:((Xr Xx) Xy)) (x30:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x400:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx0) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx0) Xz0))))=> (((x30 x0) Xy) x3)) as proof of ((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx0) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx0) Xz0)))->((Xp x0) Xy)))
% Found (fun (Xy:a) (x3:((Xr Xx) Xy)) (x30:(forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))) (x400:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx0) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx0) Xz0))))=> (((x30 x0) Xy) x3)) as proof of (((Xr Xx) Xy)->((forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp Xx0) Xy0)) ((Xp Xy0) Xz0))->((Xp Xx0) Xz0)))->((Xp x0) Xy))))
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))):(((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx)))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))):(((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0)))))
% Found (eq_ref0 (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) b)
% Found ((eq_ref Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) as proof of (((eq Prop) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0))))) b)
% Found x8:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x5) Xz)))
% Instantiate: x9:=x5:a
% Found x8 as proof of (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x9) Xz)))
% Found x3:((Xr Xx) Xy)
% Instantiate: Xy0:=Xx:a
% Found x3 as proof of ((Xr Xy0) Xy)
% Found (x400 x3) as proof of ((Xp0 Xy0) Xy)
% Found ((x40 Xy) x3) as proof of ((Xp0 Xy0) Xy)
% Found (((x4 Xy0) Xy) x3) as proof of ((Xp0 Xy0) Xy)
% Found (((x4 Xy0) Xy) x3) as proof of ((Xp0 Xy0) Xy)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))):(((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (fun (x:a)=> ((and ((Xr Xx) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xz0:a)=> ((and ((Xr Xy) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))):(((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) (fun (x:a)=> ((and ((Xr Xx) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz))))))
% Found (eta_expansion_dep00 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) as proof of (((eq (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) b)
% Found x7:((Xr Xx) x5)
% Instantiate: x9:=x5:a
% Found x7 as proof of ((Xr Xx) x9)
% Found eq_ref00:=(eq_ref0 (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))):(((eq Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy))))
% Found (eq_ref0 (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) as proof of (((eq Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) as proof of (((eq Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) as proof of (((eq Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) as proof of (((eq Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x2) Xy)))) b)
% Found x9:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x6) Xz)))
% Instantiate: x3:=x6:a
% Found x9 as proof of (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x3) Xz)))
% Found eq_ref00:=(eq_ref0 Xt):(((eq a) Xt) Xt)
% Found (eq_ref0 Xt) as proof of (((eq a) Xt) b)
% Found ((eq_ref a) Xt) as proof of (((eq a) Xt) b)
% Found ((eq_ref a) Xt) as proof of (((eq a) Xt) b)
% Found ((eq_ref a) Xt) as proof of (((eq a) Xt) b)
% Found x9:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x6) Xz)))
% Instantiate: x5:=x6:a
% Found x9 as proof of (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x5) Xz)))
% Found x9:(forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x5) Xz)))
% Instantiate: x7:=x5:a
% Found x9 as proof of (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))
% Found x8:((Xr Xx) x6)
% Instantiate: x3:=x6:a
% Found x8 as proof of ((Xr Xx) x3)
% Found x3:((Xr Xx) Xy)
% Instantiate: Xy0:=Xx:a
% Found x3 as proof of ((Xr Xy0) Xy)
% Found (x400 x3) as proof of ((Xp0 Xy0) Xy)
% Found ((x40 Xy) x3) as proof of ((Xp0 Xy0) Xy)
% Found (((x4 Xy0) Xy) x3) as proof of ((Xp0 Xy0) Xy)
% Found (((x4 Xy0) Xy) x3) as proof of ((Xp0 Xy0) Xy)
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xx) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz))))))
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xx) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz))))))
% Found x8:((Xr Xx) x5)
% Instantiate: x7:=x5:a
% Found x8 as proof of ((Xr Xx) x7)
% Found x8:((Xr Xx) x6)
% Instantiate: x5:=x6:a
% Found x8 as proof of ((Xr Xx) x5)
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xx) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz))))))
% Found eq_ref00:=(eq_ref0 (f x7)):(((eq Prop) (f x7)) (f x7))
% Found (eq_ref0 (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found ((eq_ref Prop) (f x7)) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (((eq Prop) (f x7)) ((and ((Xr Xx) x7)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x7) Xz)))))
% Found (fun (x7:a)=> ((eq_ref Prop) (f x7))) as proof of (forall (x:a), (((eq Prop) (f x)) ((and ((Xr Xx) x)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x) Xz))))))
% Found eq_ref00:=(eq_ref0 (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))):(((eq Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy))))
% Found (eq_ref0 (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) as proof of (((eq Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) as proof of (((eq Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) as proof of (((eq Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) b)
% Found ((eq_ref Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) as proof of (((eq Prop) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz0))->((Xp0 Xx0) Xz0))))->((Xp0 x0) Xy)))) b)
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found eq_substitution:=(fun (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)) (H:(((eq T) a) b))=> ((H (fun (x:T)=> (((eq U) (f a)) (f x)))) ((eq_ref U) (f a)))):(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b))))
% Instantiate: b:=(forall (T:Type) (U:Type) (a:T) (b:T) (f:(T->U)), ((((eq T) a) b)->(((eq U) (f a)) (f b)))):Prop
% Found eq_substitution as proof of b
% Found x2:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt)))
% Instantiate: x0:=Xs:a
% Found (fun (x2:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt))))=> x2) as proof of b
% Found (fun (x1:(not (((eq a) Xs) Xt))) (x2:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt))))=> x2) as proof of ((forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt)))->b)
% Found (fun (x1:(not (((eq a) Xs) Xt))) (x2:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt))))=> x2) as proof of ((not (((eq a) Xs) Xt))->((forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt)))->b))
% Found (and_rect00 (fun (x1:(not (((eq a) Xs) Xt))) (x2:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt))))=> x2)) as proof of b
% Found ((and_rect0 b) (fun (x1:(not (((eq a) Xs) Xt))) (x2:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt))))=> x2)) as proof of b
% Found (((fun (P0:Type) (x1:((not (((eq a) Xs) Xt))->((forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt)))->P0)))=> (((((and_rect (not (((eq a) Xs) Xt))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt)))) P0) x1) x)) b) (fun (x1:(not (((eq a) Xs) Xt))) (x2:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt))))=> x2)) as proof of b
% Found (((fun (P0:Type) (x1:((not (((eq a) Xs) Xt))->((forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt)))->P0)))=> (((((and_rect (not (((eq a) Xs) Xt))) (forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt)))) P0) x1) x)) b) (fun (x1:(not (((eq a) Xs) Xt))) (x2:(forall (Xp:(a->(a->Prop))), (((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xp Xx) Xy)))) (forall (Xx:a) (Xy:a) (Xz:a), (((and ((Xp Xx) Xy)) ((Xp Xy) Xz))->((Xp Xx) Xz))))->((Xp Xs) Xt))))=> x2)) as proof of b
% Found x20:=(x2 (fun (x5:a) (x40:a)=> ((Xr x5) x0))):(((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xr Xx) x0)))) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0)))))->((Xr Xs) x0))
% Instantiate: b:=(((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xr Xx) x0)))) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0)))))->((Xr Xs) x0)):Prop
% Found x20 as proof of b
% Found x6:((Xr Xx) Xy)
% Instantiate: x2:=Xy:a
% Found (fun (x6:((Xr Xx) Xy))=> x6) as proof of ((Xr Xx) x2)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))->(P0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found ((eq_ref0 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found (((eq_ref (a->Prop)) (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz)))))) P0) as proof of (P1 (fun (Xz0:a)=> ((and ((Xr Xx) Xz0)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 Xz0) Xz))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) b)->((ex a) b))
% Found (eq_ref00 (ex a)) as proof of (((ex a) b)->((ex a) b))
% Found ((eq_ref0 b) (ex a)) as proof of (((ex a) b)->((ex a) b))
% Found (((eq_ref (a->Prop)) b) (ex a)) as proof of (((ex a) b)->((ex a) b))
% Found (fun (x3:((ex a) b))=> (((eq_ref (a->Prop)) b) (ex a))) as proof of (((ex a) b)->((ex a) b))
% Found (fun (x3:((ex a) b))=> (((eq_ref (a->Prop)) b) (ex a))) as proof of (((ex a) b)->(((ex a) b)->((ex a) b)))
% Found (and_rect10 (fun (x3:((ex a) b))=> (((eq_ref (a->Prop)) b) (ex a)))) as proof of ((ex a) b)
% Found ((and_rect1 ((ex a) b)) (fun (x3:((ex a) b))=> (((eq_ref (a->Prop)) b) (ex a)))) as proof of ((ex a) b)
% Found (((fun (P0:Type) (x3:(((ex a) b)->(((ex a) b)->P0)))=> (((((and_rect ((ex a) b)) ((ex a) b)) P0) x3) x2)) ((ex a) b)) (fun (x3:((ex a) b))=> (((eq_ref (a->Prop)) b) (ex a)))) as proof of ((ex a) b)
% Found (fun (x2:((and ((ex a) b)) ((ex a) b)))=> (((fun (P0:Type) (x3:(((ex a) b)->(((ex a) b)->P0)))=> (((((and_rect ((ex a) b)) ((ex a) b)) P0) x3) x2)) ((ex a) b)) (fun (x3:((ex a) b))=> (((eq_ref (a->Prop)) b) (ex a))))) as proof of ((ex a) b)
% Found (fun (Xz:a) (x2:((and ((ex a) b)) ((ex a) b)))=> (((fun (P0:Type) (x3:(((ex a) b)->(((ex a) b)->P0)))=> (((((and_rect ((ex a) b)) ((ex a) b)) P0) x3) x2)) ((ex a) b)) (fun (x3:((ex a) b))=> (((eq_ref (a->Prop)) b) (ex a))))) as proof of (((and ((ex a) b)) ((ex a) b))->((ex a) b))
% Found (fun (Xy:a) (Xz:a) (x2:((and ((ex a) b)) ((ex a) b)))=> (((fun (P0:Type) (x3:(((ex a) b)->(((ex a) b)->P0)))=> (((((and_rect ((ex a) b)) ((ex a) b)) P0) x3) x2)) ((ex a) b)) (fun (x3:((ex a) b))=> (((eq_ref (a->Prop)) b) (ex a))))) as proof of (a->(((and ((ex a) b)) ((ex a) b))->((ex a) b)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and ((ex a) b)) ((ex a) b)))=> (((fun (P0:Type) (x3:(((ex a) b)->(((ex a) b)->P0)))=> (((((and_rect ((ex a) b)) ((ex a) b)) P0) x3) x2)) ((ex a) b)) (fun (x3:((ex a) b))=> (((eq_ref (a->Prop)) b) (ex a))))) as proof of (a->(a->(((and ((ex a) b)) ((ex a) b))->((ex a) b))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x2:((and ((ex a) b)) ((ex a) b)))=> (((fun (P0:Type) (x3:(((ex a) b)->(((ex a) b)->P0)))=> (((((and_rect ((ex a) b)) ((ex a) b)) P0) x3) x2)) ((ex a) b)) (fun (x3:((ex a) b))=> (((eq_ref (a->Prop)) b) (ex a))))) as proof of (a->(a->(a->(((and ((ex a) b)) ((ex a) b))->((ex a) b)))))
% Found x6:((Xr Xx) Xy)
% Instantiate: x2:=Xy:a
% Found (fun (x6:((Xr Xx) Xy))=> x6) as proof of ((Xr Xx) x2)
% Found eq_ref000:=(eq_ref00 (Xr Xx)):(((Xr Xx) Xy)->((Xr Xx) Xy))
% Found (eq_ref00 (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found ((eq_ref0 Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found (((eq_ref a) Xy) (Xr Xx)) as proof of (((Xr Xx) Xy)->((Xr Xx) x2))
% Found x20:=(x2 (fun (x5:a) (x40:a)=> ((Xr x5) x0))):(((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xr Xx) x0)))) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0)))))->((Xr Xs) x0))
% Instantiate: b:=(((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xr Xx) x0)))) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) x0)) ((Xr Xy) x0))->((Xr Xx) x0)))))->((Xr Xs) x0)):Prop
% Found x20 as proof of b
% Found x20:=(x2 (fun (x5:a) (x40:a)=> ((Xr x5) x5))):(((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xr Xx) Xx)))) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx)))))->((Xr Xs) Xs))
% Instantiate: b:=(((and (forall (Xx:a) (Xy:a), (((Xr Xx) Xy)->((Xr Xx) Xx)))) (forall (Xx:a) (Xy:a), (a->(((and ((Xr Xx) Xx)) ((Xr Xy) Xy))->((Xr Xx) Xx)))))->((Xr Xs) Xs)):Prop
% Found x20 as proof of b
% Found eq_ref000:=(eq_ref00 (Xp x0)):(((Xp x0) Xz)->((Xp x0) Xz))
% Found (eq_ref00 (Xp x0)) as proof of (((Xp x0) Xz)->((Xp x0) Xz))
% Found ((eq_ref0 Xz) (Xp x0)) as proof of (((Xp x0) Xz)->((Xp x0) Xz))
% Found (((eq_ref a) Xz) (Xp x0)) as proof of (((Xp x0) Xz)->((Xp x0) Xz))
% Found (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))) as proof of (((Xp x0) Xz)->((Xp x0) Xz))
% Found (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))) as proof of (((Xp x0) Xy)->(((Xp x0) Xz)->((Xp x0) Xz)))
% Found (and_rect20 (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0)))) as proof of ((Xp x0) Xz)
% Found ((and_rect2 ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0)))) as proof of ((Xp x0) Xz)
% Found (((fun (P:Type) (x6:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x6) x5)) ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0)))) as proof of ((Xp x0) Xz)
% Found (fun (x5:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x6:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x6) x5)) ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of ((Xp x0) Xz)
% Found (fun (Xz:a) (x5:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x6:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x6) x5)) ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found (fun (Xy:a) (Xz:a) (x5:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x6:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x6) x5)) ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of (forall (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x5:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x6:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x6) x5)) ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of (forall (Xy:a) (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x5:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x6:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x6) x5)) ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of (a->(forall (Xy:a) (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))))
% Found eq_ref000:=(eq_ref00 (Xp x0)):(((Xp x0) Xz)->((Xp x0) Xz))
% Found (eq_ref00 (Xp x0)) as proof of (((Xp x0) Xz)->((Xp x0) Xz))
% Found ((eq_ref0 Xz) (Xp x0)) as proof of (((Xp x0) Xz)->((Xp x0) Xz))
% Found (((eq_ref a) Xz) (Xp x0)) as proof of (((Xp x0) Xz)->((Xp x0) Xz))
% Found (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))) as proof of (((Xp x0) Xz)->((Xp x0) Xz))
% Found (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))) as proof of (((Xp x0) Xy)->(((Xp x0) Xz)->((Xp x0) Xz)))
% Found (and_rect20 (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0)))) as proof of ((Xp x0) Xz)
% Found ((and_rect2 ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0)))) as proof of ((Xp x0) Xz)
% Found (((fun (P:Type) (x6:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x6) x5)) ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0)))) as proof of ((Xp x0) Xz)
% Found (fun (x5:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x6:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x6) x5)) ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of ((Xp x0) Xz)
% Found (fun (Xz:a) (x5:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x6:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x6) x5)) ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))
% Found (fun (Xy:a) (Xz:a) (x5:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x6:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x6) x5)) ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of (forall (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x5:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x6:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x6) x5)) ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of (forall (Xy:a) (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz)))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x5:((and ((Xp x0) Xy)) ((Xp x0) Xz)))=> (((fun (P:Type) (x6:(((Xp x0) Xy)->(((Xp x0) Xz)->P)))=> (((((and_rect ((Xp x0) Xy)) ((Xp x0) Xz)) P) x6) x5)) ((Xp x0) Xz)) (fun (x6:((Xp x0) Xy))=> (((eq_ref a) Xz) (Xp x0))))) as proof of (a->(forall (Xy:a) (Xz:a), (((and ((Xp x0) Xy)) ((Xp x0) Xz))->((Xp x0) Xz))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))):(((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found (eq_ref0 ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))):(((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found (eq_ref0 ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))):(((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found (eq_ref0 ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xx) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy:a), (((Xr Xx0) Xy)->((Xp0 Xx0) Xy)))) (forall (Xx0:a) (Xy:a) (Xz00:a), (((and ((Xp0 Xx0) Xy)) ((Xp0 Xy) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))):(((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz)))))
% Found (eq_ref0 ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp0 Xx0) Xz00))))->((Xp0 x4) Xz))))) as proof of (((eq Prop) ((and ((Xr Xy) x4)) (forall (Xp0:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((Xr Xx0) Xy0)->((Xp0 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz00:a), (((and ((Xp0 Xx0) Xy0)) ((Xp0 Xy0) Xz00))->((Xp
% EOF
%------------------------------------------------------------------------------