%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV132^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n092.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 13:33:47 EDT 2014

% Result   : Timeout 300.05s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV132^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n092.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 08:11:46 CDT 2014
% % CPUTime  : 300.05 
% 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 0x1dc9830>, <kernel.Type object at 0x1dc9d40>) 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 (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xs)->(Xx Xt)))))->((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))))) of role conjecture named cTC_INTERP_BBP_OLD_pme
% Conjecture to prove = (forall (Xr:(a->(a->Prop))) (Xs:a) (Xt:a), (((and (not (((eq a) Xs) Xt))) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xs)->(Xx Xt)))))->((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx 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 (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xs)->(Xx Xt)))))->((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt)))))))))']
% Parameter a:Type.
% Trying to prove (forall (Xr:(a->(a->Prop))) (Xs:a) (Xt:a), (((and (not (((eq a) Xs) Xt))) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xs)->(Xx Xt)))))->((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt)))))))))
% Found x1:(forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xs)->(Xx Xt))))
% Instantiate: x2:=Xs:a
% Found x1 as proof of (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))
% Found x000:(Xx x0)
% Instantiate: x0:=Xt:a
% Found (fun (x000:(Xx x0))=> x000) as proof of (Xx Xt)
% Found (fun (x00:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (x000:(Xx x0))=> x000) as proof of ((Xx x0)->(Xx Xt))
% Found (fun (Xx:(a->Prop)) (x00:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (x000:(Xx x0))=> x000) as proof of ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))
% Found (fun (Xx:(a->Prop)) (x00:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (x000:(Xx x0))=> x000) as proof of (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))
% Found x2:(forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xs)->(Xx Xt))))
% Instantiate: x0:=Xs:a
% Found x2 as proof of (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))
% Found eq_ref00:=(eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))):(((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt)))))))
% Found (eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) b)
% Found eq_ref000:=(eq_ref00 Xx0):((Xx0 x2)->(Xx0 x2))
% Found (eq_ref00 Xx0) as proof of ((Xx0 x2)->(Xx0 Xs))
% Found ((eq_ref0 x2) Xx0) as proof of ((Xx0 x2)->(Xx0 Xs))
% Found (((eq_ref a) x2) Xx0) as proof of ((Xx0 x2)->(Xx0 Xs))
% Found (((eq_ref a) x2) Xx0) as proof of ((Xx0 x2)->(Xx0 Xs))
% Found (fun (x00:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz))))=> (((eq_ref a) x2) Xx0)) as proof of ((Xx0 x2)->(Xx0 Xs))
% Found (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz))))=> (((eq_ref a) x2) Xx0)) as proof of ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs)))
% Found (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz))))=> (((eq_ref a) x2) Xx0)) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))):(((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) (fun (x:a)=> ((and ((Xr Xs) x)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x)->(Xx Xt)))))))
% Found (eta_expansion_dep00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) b)
% Found ((eta_expansion_dep0 (fun (x3:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt))))))) b)
% Found x000:(Xx0 x0)
% Instantiate: x0:=Xs:a
% Found (fun (x000:(Xx0 x0))=> x000) as proof of (Xx0 Xs)
% Found (fun (x00:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))) (x000:(Xx0 x0))=> x000) as proof of ((Xx0 x0)->(Xx0 Xs))
% Found (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))) (x000:(Xx0 x0))=> x000) as proof of ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xs)))
% Found (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))) (x000:(Xx0 x0))=> x000) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xs))))
% Found x6:(Xx0 x2)
% Instantiate: x2:=Xs:a
% Found (fun (x6:(Xx0 x2))=> x6) as proof of (Xx0 Xs)
% Found (fun (x5:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))) (x6:(Xx0 x2))=> x6) as proof of ((Xx0 x2)->(Xx0 Xs))
% Found (fun (Xx0:(a->Prop)) (x5:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))) (x6:(Xx0 x2))=> x6) as proof of ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs)))
% Found (fun (Xx0:(a->Prop)) (x5:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))) (x6:(Xx0 x2))=> x6) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs))))
% Found x01:(Xx0 x2)
% Instantiate: x2:=Xz:a
% Found (fun (x01:(Xx0 x2))=> x01) as proof of (Xx0 Xz)
% Found (fun (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01) as proof of ((Xx0 x2)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 (ex a)):(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->((ex a) (fun (x:a)=> ((and ((Xr Xs) x)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x)->(Xx0 Xy))))))))
% Found (eta_expansion_dep000 (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found ((eta_expansion_dep00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found eq_ref000:=(eq_ref00 Xx0):((Xx0 x0)->(Xx0 x0))
% Found (eq_ref00 Xx0) as proof of ((Xx0 x0)->(Xx0 Xz))
% Found ((eq_ref0 x0) Xx0) as proof of ((Xx0 x0)->(Xx0 Xz))
% Found (((eq_ref a) x0) Xx0) as proof of ((Xx0 x0)->(Xx0 Xz))
% Found (((eq_ref a) x0) Xx0) as proof of ((Xx0 x0)->(Xx0 Xz))
% Found (fun (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0)) as proof of ((Xx0 x0)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0)) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0)) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found ((eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found ((eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found eq_ref000:=(eq_ref00 Xx0):((Xx0 x3)->(Xx0 x3))
% Found (eq_ref00 Xx0) as proof of ((Xx0 x3)->(Xx0 Xz))
% Found ((eq_ref0 x3) Xx0) as proof of ((Xx0 x3)->(Xx0 Xz))
% Found (((eq_ref a) x3) Xx0) as proof of ((Xx0 x3)->(Xx0 Xz))
% Found (((eq_ref a) x3) Xx0) as proof of ((Xx0 x3)->(Xx0 Xz))
% Found (fun (x5:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x3) Xx0)) as proof of ((Xx0 x3)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x5:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x3) Xx0)) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x3)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x5:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x3) Xx0)) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x3)->(Xx0 Xz))))
% Found eq_ref00:=(eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs))))))):(((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs))))))) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs)))))))
% Found (eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs))))))) b)
% Found eta_expansion000:=(eta_expansion00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))):(((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) (fun (x:a)=> ((and ((Xr Xs) x)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x)->(Xx0 Xz)))))))
% Found (eta_expansion00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found x01:(Xx0 x2)
% Instantiate: x2:=Xz:a
% Found (fun (x01:(Xx0 x2))=> x01) as proof of (Xx0 Xz)
% Found (fun (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01) as proof of ((Xx0 x2)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))
% Found eq_ref000:=(eq_ref00 Xx0):((Xx0 x0)->(Xx0 x0))
% Found (eq_ref00 Xx0) as proof of ((Xx0 x0)->(Xx0 Xz))
% Found ((eq_ref0 x0) Xx0) as proof of ((Xx0 x0)->(Xx0 Xz))
% Found (((eq_ref a) x0) Xx0) as proof of ((Xx0 x0)->(Xx0 Xz))
% Found (((eq_ref a) x0) Xx0) as proof of ((Xx0 x0)->(Xx0 Xz))
% Found (fun (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0)) as proof of ((Xx0 x0)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0)) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0)) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))
% Found x6:((Xr Xs) x2)
% Found (fun (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6) as proof of ((Xr Xs) x2)
% Found (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6) as proof of ((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->((Xr Xs) x2))
% Found (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6) as proof of (((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->((Xr Xs) x2)))
% Found (and_rect20 (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x2)
% Found ((and_rect2 ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x2)
% Found (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x2)
% Found (fun (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))) as proof of ((Xr Xs) x2)
% Found (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))) as proof of (((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))->((Xr Xs) x2))
% Found (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))) as proof of (((Xr Xy) Xz)->(((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))->((Xr Xs) x2)))
% Found (and_rect10 (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6)))) as proof of ((Xr Xs) x2)
% Found ((and_rect1 ((Xr Xs) x2)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6)))) as proof of ((Xr Xs) x2)
% Found (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))) P) x4) x3)) ((Xr Xs) x2)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6)))) as proof of ((Xr Xs) x2)
% Found (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))) P) x4) x3)) ((Xr Xs) x2)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6)))) as proof of ((Xr Xs) x2)
% Found ((conj00 (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))) P) x4) x3)) ((Xr Xs) x2)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01)) as proof of ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))
% Found (((conj0 (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))) (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))) P) x4) x3)) ((Xr Xs) x2)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01)) as proof of ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))
% Found ((((conj ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))) (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))) P) x4) x3)) ((Xr Xs) x2)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01)) as proof of ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))
% Found (fun (x3:((and ((Xr Xy) Xz)) ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz:a), (((and ((Xr Xy0) Xz)) (Xx0 Xy0))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xy)))))))=> ((((conj ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))) (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))) P) x4) x3)) ((Xr Xs) x2)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01))) as proof of ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))
% Found x6:((Xr Xs) x2)
% Found (fun (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6) as proof of ((Xr Xs) x2)
% Found (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6) as proof of ((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->((Xr Xs) x2))
% Found (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6) as proof of (((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->((Xr Xs) x2)))
% Found (and_rect20 (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x2)
% Found ((and_rect2 ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x2)
% Found (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x2)
% Found (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x2)
% Found ((conj00 (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01)) as proof of ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))
% Found (((conj0 (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01)) as proof of ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))
% Found ((((conj ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01)) as proof of ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))
% Found (fun (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01))) as proof of ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))
% Found (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01))) as proof of (((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))->((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))))
% Found (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01))) as proof of (((Xr Xy) Xz)->(((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))->((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))))
% Found (and_rect10 (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01)))) as proof of ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))
% Found ((and_rect1 ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01)))) as proof of ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))
% Found (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))) P) x4) x3)) ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01)))) as proof of ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))
% Found (fun (x3:((and ((Xr Xy) Xz)) ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz:a), (((and ((Xr Xy0) Xz)) (Xx0 Xy0))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xy)))))))=> (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))) P) x4) x3)) ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x2)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x2)) (fun (x6:((Xr Xs) x2)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))) (x01:(Xx0 x2))=> x01))))) as proof of ((and ((Xr Xs) x2)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x2)->(Xx0 Xz)))))
% Found eq_ref000:=(eq_ref00 Xx0):((Xx0 x5)->(Xx0 x5))
% Found (eq_ref00 Xx0) as proof of ((Xx0 x5)->(Xx0 Xz))
% Found ((eq_ref0 x5) Xx0) as proof of ((Xx0 x5)->(Xx0 Xz))
% Found (((eq_ref a) x5) Xx0) as proof of ((Xx0 x5)->(Xx0 Xz))
% Found (((eq_ref a) x5) Xx0) as proof of ((Xx0 x5)->(Xx0 Xz))
% Found (fun (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x5) Xx0)) as proof of ((Xx0 x5)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x5) Xx0)) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x5) Xx0)) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xz))))
% Found x6:((Xr Xs) x0)
% Found (fun (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6) as proof of ((Xr Xs) x0)
% Found (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6) as proof of ((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->((Xr Xs) x0))
% Found (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6) as proof of (((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->((Xr Xs) x0)))
% Found (and_rect20 (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x0)
% Found ((and_rect2 ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x0)
% Found (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x0)
% Found (fun (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))) as proof of ((Xr Xs) x0)
% Found (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))) as proof of (((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))->((Xr Xs) x0))
% Found (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))) as proof of (((Xr Xy) Xz)->(((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))->((Xr Xs) x0)))
% Found (and_rect10 (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6)))) as proof of ((Xr Xs) x0)
% Found ((and_rect1 ((Xr Xs) x0)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6)))) as proof of ((Xr Xs) x0)
% Found (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))) P) x4) x3)) ((Xr Xs) x0)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6)))) as proof of ((Xr Xs) x0)
% Found (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))) P) x4) x3)) ((Xr Xs) x0)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6)))) as proof of ((Xr Xs) x0)
% Found ((conj00 (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))) P) x4) x3)) ((Xr Xs) x0)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0))) as proof of ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))
% Found (((conj0 (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))) (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))) P) x4) x3)) ((Xr Xs) x0)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0))) as proof of ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))
% Found ((((conj ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))) (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))) P) x4) x3)) ((Xr Xs) x0)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0))) as proof of ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))
% Found (fun (x3:((and ((Xr Xy) Xz)) ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz:a), (((and ((Xr Xy0) Xz)) (Xx0 Xy0))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xy)))))))=> ((((conj ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))) (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))) P) x4) x3)) ((Xr Xs) x0)) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0)))) as proof of ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))
% Found x6:((Xr Xs) x0)
% Found (fun (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6) as proof of ((Xr Xs) x0)
% Found (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6) as proof of ((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->((Xr Xs) x0))
% Found (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6) as proof of (((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->((Xr Xs) x0)))
% Found (and_rect20 (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x0)
% Found ((and_rect2 ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x0)
% Found (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x0)
% Found (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6)) as proof of ((Xr Xs) x0)
% Found ((conj00 (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0))) as proof of ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))
% Found (((conj0 (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0))) as proof of ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))
% Found ((((conj ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0))) as proof of ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))
% Found (fun (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0)))) as proof of ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))
% Found (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0)))) as proof of (((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))->((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))))
% Found (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0)))) as proof of (((Xr Xy) Xz)->(((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))->((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))))
% Found (and_rect10 (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0))))) as proof of ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))
% Found ((and_rect1 ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0))))) as proof of ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))
% Found (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))) P) x4) x3)) ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0))))) as proof of ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))
% Found (fun (x3:((and ((Xr Xy) Xz)) ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz:a), (((and ((Xr Xy0) Xz)) (Xx0 Xy0))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xy)))))))=> (((fun (P:Type) (x4:(((Xr Xy) Xz)->(((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))->P)))=> (((((and_rect ((Xr Xy) Xz)) ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))) P) x4) x3)) ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))) (fun (x4:((Xr Xy) Xz)) (x5:((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))))=> ((((conj ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz))))) (((fun (P:Type) (x6:(((Xr Xs) x0)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy))))) P) x6) x5)) ((Xr Xs) x0)) (fun (x6:((Xr Xs) x0)) (x7:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xy)))))=> x6))) (fun (Xx0:(a->Prop)) (x00:(forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0))))=> (((eq_ref a) x0) Xx0)))))) as proof of ((and ((Xr Xs) x0)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x0)->(Xx0 Xz)))))
% Found eq_ref000:=(eq_ref00 Xx0):((Xx0 x3)->(Xx0 x3))
% Found (eq_ref00 Xx0) as proof of ((Xx0 x3)->(Xx0 Xz))
% Found ((eq_ref0 x3) Xx0) as proof of ((Xx0 x3)->(Xx0 Xz))
% Found (((eq_ref a) x3) Xx0) as proof of ((Xx0 x3)->(Xx0 Xz))
% Found (((eq_ref a) x3) Xx0) as proof of ((Xx0 x3)->(Xx0 Xz))
% Found (fun (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x3) Xx0)) as proof of ((Xx0 x3)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x3) Xx0)) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x3)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x3) Xx0)) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x3)->(Xx0 Xz))))
% Found x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Instantiate: b:=(fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))):(a->Prop)
% Found x4 as proof of (P b)
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))):(((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) (fun (x:a)=> ((and ((Xr Xs) x)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x)->(Xx0 Xz)))))))
% Found (eta_expansion_dep00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found ((eta_expansion_dep0 (fun (x6:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found (((eta_expansion_dep a) (fun (x6:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))))
% Found (eq_ref00 (ex a)) as proof of (((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P b))
% Found ((eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P b))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P b))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P b))
% Found (fun (x3:((Xr Xy) Xz))=> (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a))) as proof of (((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P b))
% Found (fun (x3:((Xr Xy) Xz))=> (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a))) as proof of (((Xr Xy) Xz)->(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P b)))
% Found (and_rect10 (fun (x3:((Xr Xy) Xz))=> (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)))) as proof of (P b)
% Found ((and_rect1 (P b)) (fun (x3:((Xr Xy) Xz))=> (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)))) as proof of (P b)
% Found (((fun (P0:Type) (x3:(((Xr Xy) Xz)->(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->P0)))=> (((((and_rect ((Xr Xy) Xz)) ((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))) P0) x3) x2)) (P b)) (fun (x3:((Xr Xy) Xz))=> (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)))) as proof of (P b)
% Found (((fun (P0:Type) (x3:(((Xr Xy) Xz)->(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->P0)))=> (((((and_rect ((Xr Xy) Xz)) ((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))) P0) x3) x2)) (P b)) (fun (x3:((Xr Xy) Xz))=> (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)))) as proof of (P b)
% Found x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Instantiate: f:=(fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))):(a->Prop)
% Found x4 as proof of (P f)
% Found x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Instantiate: f:=(fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))):(a->Prop)
% Found x4 as proof of (P f)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))):(((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (fun (x:a)=> ((and ((Xr Xs) x)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x)->(Xx0 Xy)))))))
% Found (eta_expansion00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found eq_ref000:=(eq_ref00 (ex a)):(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))))
% Found (eq_ref00 (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found ((eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (P (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found eta_expansion000:=(eta_expansion00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))):(((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (fun (x:a)=> ((and ((Xr Xs) x)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x)->(Xx0 Xy)))))))
% Found (eta_expansion00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt)))))))
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))):(((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) b)
% Found x60:(Xx0 x7)
% Instantiate: x7:=Xz:a
% Found (fun (x60:(Xx0 x7))=> x60) as proof of (Xx0 Xz)
% Found (fun (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x60:(Xx0 x7))=> x60) as proof of ((Xx0 x7)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x60:(Xx0 x7))=> x60) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x7)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x60:(Xx0 x7))=> x60) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x7)->(Xx0 Xz))))
% Found x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Instantiate: f:=(fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))):(a->Prop)
% Found (fun (x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))))=> x4) as proof of (P f)
% Found (fun (x3:((Xr Xy) Xz)) (x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))))=> x4) as proof of (((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P f))
% Found (fun (x3:((Xr Xy) Xz)) (x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))))=> x4) as proof of (((Xr Xy) Xz)->(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P f)))
% Found (and_rect10 (fun (x3:((Xr Xy) Xz)) (x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))))=> x4)) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:((Xr Xy) Xz)) (x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))))=> x4)) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((Xr Xy) Xz)->(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->P0)))=> (((((and_rect ((Xr Xy) Xz)) ((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))) P0) x3) x2)) (P f)) (fun (x3:((Xr Xy) Xz)) (x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))))=> x4)) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((Xr Xy) Xz)->(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->P0)))=> (((((and_rect ((Xr Xy) Xz)) ((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))) P0) x3) x2)) (P f)) (fun (x3:((Xr Xy) Xz)) (x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))))=> x4)) as proof of (P f)
% Found eta_expansion0000:=(eta_expansion000 (ex a)):(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->((ex a) (fun (x:a)=> ((and ((Xr Xs) x)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x)->(Xx0 Xy))))))))
% Found (eta_expansion000 (ex a)) as proof of (((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P f))
% Found ((eta_expansion00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P f))
% Found (((eta_expansion0 Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P f))
% Found ((((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P f))
% Found ((((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)) as proof of (((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P f))
% Found (fun (x3:((Xr Xy) Xz))=> ((((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a))) as proof of (((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P f))
% Found (fun (x3:((Xr Xy) Xz))=> ((((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a))) as proof of (((Xr Xy) Xz)->(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->(P f)))
% Found (and_rect10 (fun (x3:((Xr Xy) Xz))=> ((((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)))) as proof of (P f)
% Found ((and_rect1 (P f)) (fun (x3:((Xr Xy) Xz))=> ((((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((Xr Xy) Xz)->(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->P0)))=> (((((and_rect ((Xr Xy) Xz)) ((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))) P0) x3) x2)) (P f)) (fun (x3:((Xr Xy) Xz))=> ((((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)))) as proof of (P f)
% Found (((fun (P0:Type) (x3:(((Xr Xy) Xz)->(((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))->P0)))=> (((((and_rect ((Xr Xy) Xz)) ((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))) P0) x3) x2)) (P f)) (fun (x3:((Xr Xy) Xz))=> ((((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (ex a)))) as proof of (P f)
% Found x60:(Xx0 x3)
% Instantiate: x3:=Xz:a
% Found (fun (x60:(Xx0 x3))=> x60) as proof of (Xx0 Xz)
% Found (fun (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x60:(Xx0 x3))=> x60) as proof of ((Xx0 x3)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x60:(Xx0 x3))=> x60) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x3)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x60:(Xx0 x3))=> x60) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x3)->(Xx0 Xz))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xc)->(Xx Xt)))))))
% 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 eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found x60:(Xx0 x5)
% Instantiate: x5:=Xz:a
% Found (fun (x60:(Xx0 x5))=> x60) as proof of (Xx0 Xz)
% Found (fun (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x60:(Xx0 x5))=> x60) as proof of ((Xx0 x5)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x60:(Xx0 x5))=> x60) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x60:(Xx0 x5))=> x60) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xz))))
% Found x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Instantiate: b:=(fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))):(a->Prop)
% Found x4 as proof of (P b)
% 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 eta_expansion000:=(eta_expansion00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))):(((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) (fun (x:a)=> ((and ((Xr Xs) x)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x)->(Xx0 Xz)))))))
% Found (eta_expansion00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found ((eta_expansion0 Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found (((eta_expansion a) Prop) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))):(((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))) 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 x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Instantiate: f:=(fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))):(a->Prop)
% Found x4 as proof of (P f)
% Found x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Instantiate: f:=(fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))):(a->Prop)
% Found x4 as proof of (P f)
% Found x7:((Xr Xs) x5)
% Instantiate: x9:=x5:a
% Found x7 as proof of ((Xr Xs) x9)
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))->(P0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found eta_expansion_dep0000:=(eta_expansion_dep000 P0):((P0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))->(P0 (fun (x:a)=> ((and ((Xr Xs) x)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x)->(Xx0 Xz))))))))
% Found (eta_expansion_dep000 P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eta_expansion_dep00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% 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 x8:((Xr Xs) x6)
% Instantiate: x3:=x6:a
% Found (fun (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of ((Xr Xs) x3)
% Found (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of ((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->((Xr Xs) x3))
% Found (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of (((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->((Xr Xs) x3)))
% Found (and_rect20 (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x3)
% Found ((and_rect2 ((Xr Xs) x3)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x3)
% Found (((fun (P:Type) (x8:(((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) P) x8) x7)) ((Xr Xs) x3)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x3)
% Found (fun (x7:((and ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))))=> (((fun (P:Type) (x8:(((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) P) x8) x7)) ((Xr Xs) x3)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8))) as proof of ((Xr Xs) x3)
% Found x8:((Xr Xs) x6)
% Instantiate: x3:=x6:a
% Found (fun (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of ((Xr Xs) x3)
% Found (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of ((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->((Xr Xs) x3))
% Found (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of (((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->((Xr Xs) x3)))
% Found (and_rect20 (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x3)
% Found ((and_rect2 ((Xr Xs) x3)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x3)
% Found (((fun (P:Type) (x8:(((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) P) x8) x7)) ((Xr Xs) x3)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x3)
% Found (fun (x7:((and ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))))=> (((fun (P:Type) (x8:(((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) P) x8) x7)) ((Xr Xs) x3)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8))) as proof of ((Xr Xs) x3)
% Found x8:((Xr Xs) x6)
% Instantiate: x3:=x6:a
% Found (fun (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of ((Xr Xs) x3)
% Found (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of ((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->((Xr Xs) x3))
% Found (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of (((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->((Xr Xs) x3)))
% Found (and_rect20 (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x3)
% Found ((and_rect2 ((Xr Xs) x3)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x3)
% Found (((fun (P:Type) (x8:(((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) P) x8) x7)) ((Xr Xs) x3)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x3)
% Found (((fun (P:Type) (x8:(((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) P) x8) x7)) ((Xr Xs) x3)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x3)
% Found x8:((Xr Xs) x6)
% Instantiate: x5:=x6:a
% Found (fun (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of ((Xr Xs) x5)
% Found (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of ((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->((Xr Xs) x5))
% Found (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of (((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->((Xr Xs) x5)))
% Found (and_rect20 (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x5)
% Found ((and_rect2 ((Xr Xs) x5)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x5)
% Found (((fun (P:Type) (x8:(((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) P) x8) x7)) ((Xr Xs) x5)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x5)
% Found (fun (x7:((and ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))))=> (((fun (P:Type) (x8:(((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) P) x8) x7)) ((Xr Xs) x5)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8))) as proof of ((Xr Xs) x5)
% Found eq_ref00:=(eq_ref0 (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))):(((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))
% Found (eq_ref0 (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) b)
% Found ((eq_ref Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) as proof of (((eq Prop) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))) b)
% Found x8:((Xr Xs) x6)
% Instantiate: x5:=x6:a
% Found (fun (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of ((Xr Xs) x5)
% Found (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of ((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->((Xr Xs) x5))
% Found (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8) as proof of (((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->((Xr Xs) x5)))
% Found (and_rect20 (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x5)
% Found ((and_rect2 ((Xr Xs) x5)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x5)
% Found (((fun (P:Type) (x8:(((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) P) x8) x7)) ((Xr Xs) x5)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x5)
% Found (((fun (P:Type) (x8:(((Xr Xs) x6)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) P) x8) x7)) ((Xr Xs) x5)) (fun (x8:((Xr Xs) x6)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x5)
% Found x8:((Xr Xs) x5)
% Instantiate: x7:=x5:a
% Found (fun (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xy)))))=> x8) as proof of ((Xr Xs) x7)
% Found (fun (x8:((Xr Xs) x5)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xy)))))=> x8) as proof of ((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xy))))->((Xr Xs) x7))
% Found (fun (x8:((Xr Xs) x5)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xy)))))=> x8) as proof of (((Xr Xs) x5)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xy))))->((Xr Xs) x7)))
% Found (and_rect20 (fun (x8:((Xr Xs) x5)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x7)
% Found ((and_rect2 ((Xr Xs) x7)) (fun (x8:((Xr Xs) x5)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x7)
% Found (((fun (P:Type) (x8:(((Xr Xs) x5)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x5)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xy))))) P) x8) x6)) ((Xr Xs) x7)) (fun (x8:((Xr Xs) x5)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x7)
% Found (((fun (P:Type) (x8:(((Xr Xs) x5)->((forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xy))))->P)))=> (((((and_rect ((Xr Xs) x5)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xy))))) P) x8) x6)) ((Xr Xs) x7)) (fun (x8:((Xr Xs) x5)) (x9:(forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xy)))))=> x8)) as proof of ((Xr Xs) x7)
% Found x8:((Xr Xs) x6)
% Instantiate: x3:=x6:a
% Found x8 as proof of ((Xr Xs) x3)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found eq_ref00:=(eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))):(((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 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 (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Found eq_ref00:=(eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))):(((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found x8:((Xr Xs) x5)
% Instantiate: x7:=x5:a
% Found x8 as proof of ((Xr Xs) x7)
% Found x8:((Xr Xs) x6)
% Instantiate: x5:=x6:a
% Found x8 as proof of ((Xr Xs) x5)
% Found eq_ref000:=(eq_ref00 (Xr Xs)):(((Xr Xs) Xy)->((Xr Xs) Xy))
% Found (eq_ref00 (Xr Xs)) as proof of (P Xy)
% Found ((eq_ref0 Xy) (Xr Xs)) as proof of (P Xy)
% Found (((eq_ref a) Xy) (Xr Xs)) as proof of (P Xy)
% Found (((eq_ref a) Xy) (Xr Xs)) as proof of (P Xy)
% 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 x000:(Xx x0)
% Instantiate: x0:=Xt:a
% Found (fun (x000:(Xx x0))=> x000) as proof of (Xx Xt)
% Found (fun (x00:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (x000:(Xx x0))=> x000) as proof of ((Xx x0)->(Xx Xt))
% Found (fun (Xx:(a->Prop)) (x00:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (x000:(Xx x0))=> x000) as proof of ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))
% Found (fun (Xx:(a->Prop)) (x00:(forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))) (x000:(Xx x0))=> x000) as proof of b
% Found x1:(forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xs)->(Xx Xt))))
% Instantiate: x2:=Xs:a
% Found x1 as proof of 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 eq_ref000:=(eq_ref00 P0):((P0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))->(P0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))->(P0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))->(P0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))->(P0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))))
% Found (eq_ref00 P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) P0) as proof of (P1 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% 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_ref000:=(eq_ref00 Xx0):((Xx0 x9)->(Xx0 x9))
% Found (eq_ref00 Xx0) as proof of ((Xx0 x9)->(Xx0 Xz))
% Found ((eq_ref0 x9) Xx0) as proof of ((Xx0 x9)->(Xx0 Xz))
% Found (((eq_ref a) x9) Xx0) as proof of ((Xx0 x9)->(Xx0 Xz))
% Found (((eq_ref a) x9) Xx0) as proof of ((Xx0 x9)->(Xx0 Xz))
% Found (fun (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x9) Xx0)) as proof of ((Xx0 x9)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x9) Xx0)) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x9)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x9) Xx0)) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x9)->(Xx0 Xz))))
% Found x2:(forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xs)->(Xx Xt))))
% Instantiate: x0:=Xs:a
% Found x2 as proof of b
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))):(((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))):(((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))):(((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))):(((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xz))))))
% 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 eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found (((eq_trans000 ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found ((((eq_trans00 ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found (((((eq_trans0 (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx 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)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found (((eq_trans000 ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found ((((eq_trans00 ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found (((((eq_trans0 (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found ((((((eq_trans Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt)))))) ((eq_ref Prop) (f x0))) ((eq_ref Prop) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))) as proof of (((eq Prop) (f x0)) ((and ((Xr Xs) x0)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x0)->(Xx Xt))))))
% Found x60:(Xx0 x3)
% Instantiate: x3:=Xz:a
% Found (fun (x60:(Xx0 x3))=> x60) as proof of (Xx0 Xz)
% Found (fun (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x60:(Xx0 x3))=> x60) as proof of ((Xx0 x3)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x60:(Xx0 x3))=> x60) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x3)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))) (x60:(Xx0 x3))=> x60) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x3)->(Xx0 Xz))))
% 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 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 Xx0):((Xx0 x5)->(Xx0 x5))
% Found (eq_ref00 Xx0) as proof of ((Xx0 x5)->(Xx0 Xz))
% Found ((eq_ref0 x5) Xx0) as proof of ((Xx0 x5)->(Xx0 Xz))
% Found (((eq_ref a) x5) Xx0) as proof of ((Xx0 x5)->(Xx0 Xz))
% Found (((eq_ref a) x5) Xx0) as proof of ((Xx0 x5)->(Xx0 Xz))
% Found (fun (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x5) Xx0)) as proof of ((Xx0 x5)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x5) Xx0)) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x5) Xx0)) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x5)->(Xx0 Xz))))
% Found eq_ref000:=(eq_ref00 Xx0):((Xx0 x7)->(Xx0 x7))
% Found (eq_ref00 Xx0) as proof of ((Xx0 x7)->(Xx0 Xz))
% Found ((eq_ref0 x7) Xx0) as proof of ((Xx0 x7)->(Xx0 Xz))
% Found (((eq_ref a) x7) Xx0) as proof of ((Xx0 x7)->(Xx0 Xz))
% Found (((eq_ref a) x7) Xx0) as proof of ((Xx0 x7)->(Xx0 Xz))
% Found (fun (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x7) Xx0)) as proof of ((Xx0 x7)->(Xx0 Xz))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x7) Xx0)) as proof of ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x7)->(Xx0 Xz)))
% Found (fun (Xx0:(a->Prop)) (x50:(forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0))))=> (((eq_ref a) x7) Xx0)) as proof of (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 x7)->(Xx0 Xz))))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x0))->(P0 (f x0)))
% Found (eq_ref00 P0) as proof of (P1 (f x0))
% Found ((eq_ref0 (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found (((eq_ref Prop) (f x0)) P0) as proof of (P1 (f x0))
% Found 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_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 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 x2:(forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx Xs)->(Xx Xt))))
% Instantiate: x0:=Xs:a
% Found x2 as proof of 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 eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 Xc)->(Xx0 Xs)))))))
% 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 (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found (((eta_expansion_dep a) (fun (x3:a)=> Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found eq_ref00:=(eq_ref0 a0):(((eq (a->Prop)) a0) a0)
% Found (eq_ref0 a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found ((eq_ref (a->Prop)) a0) as proof of (((eq (a->Prop)) a0) b)
% Found x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Instantiate: b:=(fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))):(a->Prop)
% Found x4 as proof of (P b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs))))):(((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs))))) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs)))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x2)->(Xx0 Xs))))) b)
% Found eq_ref00:=(eq_ref0 (f x2)):(((eq Prop) (f x2)) (f x2))
% Found (eq_ref0 (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found (((eq_trans000 ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found ((((eq_trans00 ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found (((((eq_trans0 (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found ((((((eq_trans Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx 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)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found ((eq_ref Prop) (f x2)) as proof of (((eq Prop) (f x2)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found (((eq_trans000 ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found ((((eq_trans00 ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found (((((eq_trans0 (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found ((((((eq_trans Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt)))))) ((eq_ref Prop) (f x2))) ((eq_ref Prop) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))) as proof of (((eq Prop) (f x2)) ((and ((Xr Xs) x2)) (forall (Xx:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx Xy))->(Xx Xz)))->((Xx x2)->(Xx Xt))))))
% Found eq_ref00:=(eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))):(((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found (eq_ref0 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz))))))) b)
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x2))->(P0 (f x2)))
% Found (eq_ref00 P0) as proof of (P1 (f x2))
% Found ((eq_ref0 (f x2)) P0) as proof of (P1 (f x2))
% Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% Found eq_ref000:=(eq_ref00 P0):((P0 (f x2))->(P0 (f x2)))
% Found (eq_ref00 P0) as proof of (P1 (f x2))
% Found ((eq_ref0 (f x2)) P0) as proof of (P1 (f x2))
% Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% Found (((eq_ref Prop) (f x2)) P0) as proof of (P1 (f x2))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))):(((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (fun (x:a)=> ((and ((Xr Xs) x)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x)->(Xx0 Xy)))))))
% Found (eta_expansion_dep00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))):(((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) (fun (x:a)=> ((and ((Xr Xs) x)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x)->(Xx0 Xy)))))))
% Found (eta_expansion_dep00 (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found ((eta_expansion_dep0 (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found (((eta_expansion_dep a) (fun (x5:a)=> Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) as proof of (((eq (a->Prop)) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy))))))) b)
% Found eq_ref00:=(eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xs))))):(((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xs))))) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xs)))))
% Found (eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xs))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xs))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xs))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xs))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xs))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xs))))) b)
% Found ((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xs))))) as proof of (((eq Prop) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz:a), (((and ((Xr Xy) Xz)) (Xx0 Xy))->(Xx0 Xz)))->((Xx0 x0)->(Xx0 Xs))))) b)
% Found eq_ref000:=(eq_ref00 (and ((Xr Xs) x6))):(((and ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))->((and ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))))
% Found (eq_ref00 (and ((Xr Xs) x6))) as proof of (P (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))
% Found ((eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) (and ((Xr Xs) x6))) as proof of (P (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))
% Found (((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) (and ((Xr Xs) x6))) as proof of (P (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))
% Found (((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) (and ((Xr Xs) x6))) as proof of (P (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))
% Found eq_ref000:=(eq_ref00 (and ((Xr Xs) x6))):(((and ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))->((and ((Xr Xs) x6)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))))
% Found (eq_ref00 (and ((Xr Xs) x6))) as proof of (P (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))
% Found ((eq_ref0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) (and ((Xr Xs) x6))) as proof of (P (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))
% Found (((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) (and ((Xr Xs) x6))) as proof of (P (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))
% Found (((eq_ref Prop) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy))))) (and ((Xr Xs) x6))) as proof of (P (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x6)->(Xx0 Xy)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found eq_ref000:=(eq_ref00 P0):((P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))->(P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (eq_ref00 P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found (((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) P0) as proof of (P1 (P0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Instantiate: f:=(fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))):(a->Prop)
% Found x4 as proof of (P f)
% Found x4:((ex a) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))))
% Instantiate: f:=(fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xy)))))):(a->Prop)
% Found x4 as proof of (P f)
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))):(((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))):(((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))):(((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))):(((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))):(((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))):(((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))):(((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found eq_ref00:=(eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))):(((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy))))))
% Found (eq_ref0 ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found ((eq_ref Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) as proof of (P0 (((eq Prop) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))) ((and ((Xr Xs) x4)) (forall (Xx0:(a->Prop)), ((forall (Xy0:a) (Xz0:a), (((and ((Xr Xy0) Xz0)) (Xx0 Xy0))->(Xx0 Xz0)))->((Xx0 x4)->(Xx0 Xy)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq (a->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 Xy))->(Xx0 Xz0)))->((Xx0 Xc)->(Xx0 Xz)))))))
% Found ((eq_ref (a->Prop)) b) as proof of (((eq (a->Prop)) b) (fun (Xc:a)=> ((and ((Xr Xs) Xc)) (forall (Xx0:(a->Prop)), ((forall (Xy:a) (Xz0:a), (((and ((Xr Xy) Xz0)) (Xx0 
% EOF
%------------------------------------------------------------------------------