%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV158^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n188.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 13:33:48 EDT 2014

% Result   : Timeout 300.03s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV158^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n188.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 08:15:41 CDT 2014
% % CPUTime  : 300.03 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula ((ex ((fofType->Prop)->((fofType->Prop)->Prop))) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) of role conjecture named cTHM120I_1_pme
% Conjecture to prove = ((ex ((fofType->Prop)->((fofType->Prop)->Prop))) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['((ex ((fofType->Prop)->((fofType->Prop)->Prop))) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))']
% Parameter fofType:Type.
% Trying to prove ((ex ((fofType->Prop)->((fofType->Prop)->Prop))) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))):(((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (eta_expansion_dep00 (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) b)
% Found ((eta_expansion_dep0 (fun (x1:((fofType->Prop)->((fofType->Prop)->Prop)))=> Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) b)
% Found (((eta_expansion_dep ((fofType->Prop)->((fofType->Prop)->Prop))) (fun (x1:((fofType->Prop)->((fofType->Prop)->Prop)))=> Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) b)
% Found (((eta_expansion_dep ((fofType->Prop)->((fofType->Prop)->Prop))) (fun (x1:((fofType->Prop)->((fofType->Prop)->Prop)))=> Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) b)
% Found (((eta_expansion_dep ((fofType->Prop)->((fofType->Prop)->Prop))) (fun (x1:((fofType->Prop)->((fofType->Prop)->Prop)))=> Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_trans0000 ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found ((eq_trans000 (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((eq_trans00 (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found ((((eq_trans0 (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) (Y Xx)) (X Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found ((and_rect0 (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (eq_sym000 (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of (((eq Prop) (X Xx)) (Y Xx))
% Found ((eq_sym00 (X Xx)) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of (((eq Prop) (X Xx)) (Y Xx))
% Found (((eq_sym0 (Y Xx)) (X Xx)) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of (((eq Prop) (X Xx)) (Y Xx))
% Found ((((eq_sym Prop) (Y Xx)) (X Xx)) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of (((eq Prop) (X Xx)) (Y Xx))
% Found x10:(P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P0 (X Xx))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found and_rect0000:=(and_rect000 (fun (x1:Prop)=> ((x Y) X))):(((x Y) X)->((x Y) X))
% Found (and_rect000 (fun (x1:Prop)=> ((x Y) X))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) b) (Y Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found ((and_rect0 (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> (((and_rect0 (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_trans0000 ((eq_ref Prop) (X Xx))) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))))) as proof of (((eq Prop) (X Xx)) (Y Xx))
% Found (((eq_trans000 (Y Xx)) ((eq_ref Prop) (X Xx))) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) (eq_trans000 (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) as proof of (((eq Prop) (X Xx)) (Y Xx))
% Found ((((eq_trans00 (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx))) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) ((eq_trans00 (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) as proof of (((eq Prop) (X Xx)) (Y Xx))
% Found (((((eq_trans0 (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx))) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (((eq_trans0 (X Xx)) (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) as proof of (((eq Prop) (X Xx)) (Y Xx))
% Found ((((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx))) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) ((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) as proof of (((eq Prop) (X Xx)) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) b)
% Found (eq_ref0 b) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((eq_ref (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((eq_ref (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((eq_ref (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found eq_ref00:=(eq_ref0 a):(((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) a) a)
% Found (eq_ref0 a) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) a) b)
% Found ((eq_ref (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) a) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) a) b)
% Found ((eq_ref (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) a) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) a) b)
% Found ((eq_ref (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) a) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) a) b)
% Found x10:(P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P0 (X Xx))
% Found eq_ref00:=(eq_ref0 (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))):(((eq Prop) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found (eq_ref0 (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))) as proof of (((eq Prop) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))) b)
% Found ((eq_ref Prop) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))) as proof of (((eq Prop) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))) b)
% Found ((eq_ref Prop) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))) as proof of (((eq Prop) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))) b)
% Found ((eq_ref Prop) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))) as proof of (((eq Prop) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((eq_trans000 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((((eq_trans0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((eq_trans000 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((((eq_trans0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_trans0000 ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found ((eq_trans000 (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((eq_trans00 (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found ((((eq_trans0 (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) (Y Xx)) (X Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found ((and_rect0 (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found ((eq_sym0000 (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((eq_sym0000 (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> ((eq_sym000 x1) (fun (x3:Prop)=> ((P (X Xx))->(P x3))))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((eq_sym00 (X Xx)) x1) (fun (x3:Prop)=> ((P (X Xx))->(P x3))))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> ((((eq_sym0 (Y Xx)) (X Xx)) x1) (fun (x3:Prop)=> ((P (X Xx))->(P x3))))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x1) (fun (x3:Prop)=> ((P (X Xx))->(P x3))))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop))=> (((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x1) (fun (x3:Prop)=> ((P (X Xx))->(P x3))))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_trans0000 ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found ((eq_trans000 (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((eq_trans00 (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found ((((eq_trans0 (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) (Y Xx)) (X Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found ((and_rect0 (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (eq_sym0000 (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (eq_sym0000 (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> ((eq_sym000 x1) P)) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((eq_sym00 (X Xx)) x1) P)) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> ((((eq_sym0 (Y Xx)) (X Xx)) x1) P)) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x1) P)) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop))=> ((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x1) P)) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_trans0000 ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found ((eq_trans000 (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((eq_trans00 (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found ((((eq_trans0 (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) (Y Xx)) (X Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found ((and_rect0 (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (eq_sym000 (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found ((eq_sym00 (X Xx)) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found (((eq_sym0 (Y Xx)) (X Xx)) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found ((((eq_sym Prop) (Y Xx)) (X Xx)) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found ((((eq_sym Prop) (Y Xx)) (X Xx)) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found x1:(P (X Xx))
% Instantiate: b:=(X Xx):Prop
% Found x1 as proof of (P0 b)
% Found x10:(P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P0 (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found x10:(P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P0 (X Xx))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (((((eq_trans0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (eq_sym000 ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_sym00 (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((eq_sym0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (((((eq_trans0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (eq_sym000 ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_sym00 (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((eq_sym0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found and_rect0000:=(and_rect000 (fun (x1:Prop)=> ((x Y) X))):(((x Y) X)->((x Y) X))
% Found (and_rect000 (fun (x1:Prop)=> ((x Y) X))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) b) (Y Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found ((and_rect0 (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> (((and_rect0 (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_trans0000 ((eq_ref Prop) (X Xx))) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found (((eq_trans000 (Y Xx)) ((eq_ref Prop) (X Xx))) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) b) (Y Xx))) (eq_trans000 (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found ((((eq_trans00 (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx))) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) ((eq_trans00 (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found (((((eq_trans0 (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx))) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (((eq_trans0 (X Xx)) (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found ((((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx))) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) ((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found ((((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx))) (((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P:Type) (x1:(((x X) Y)->(((x Y) X)->P)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) ((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found x1:(P (X Xx))
% Instantiate: b:=(X Xx):Prop
% Found x1 as proof of (P0 b)
% Found x10:(P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P0 (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found x10:(P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P0 (X Xx))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found eq_trans00010:=(eq_trans0001 (Y Xx)):((((eq Prop) b) (Y Xx))->(((eq Prop) (X Xx)) (Y Xx)))
% Found (eq_trans0001 (Y Xx)) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx)) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx)) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) b) (Y Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx)))) as proof of (((eq Prop) b) (Y Xx))
% Found ((and_rect0 (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx)))) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx)))) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx)))) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_trans00000 ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((eq_trans00000 ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (X Xx)) b)) (x2:(((eq Prop) b) (Y Xx)))=> (((eq_trans0000 x1) x2) P)) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (X Xx)) b)) (x2:(((eq Prop) b) (Y Xx)))=> ((((eq_trans000 (Y Xx)) x1) x2) P)) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((eq_trans00 (X Xx)) (Y Xx)) x1) x2) P)) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> (((eq_trans00 (X Xx)) c) x00)) (Y Xx))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> ((((((eq_trans0 (X Xx)) (X Xx)) (Y Xx)) x1) x2) P)) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((((eq_trans0 (X Xx)) (X Xx)) c) x00)) (Y Xx))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) x1) x2) P)) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> (((((eq_trans Prop) (X Xx)) (X Xx)) c) x00)) (Y Xx))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop))=> (((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) x1) x2) P)) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> (((((eq_trans Prop) (X Xx)) (X Xx)) c) x00)) (Y Xx)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found and_rect0000:=(and_rect000 (fun (x1:Prop)=> ((x Y) X))):(((x Y) X)->((x Y) X))
% Found (and_rect000 (fun (x1:Prop)=> ((x Y) X))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) b) (Y Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found ((and_rect0 (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> (((and_rect0 (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found (((eq_trans00000 ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((eq_trans00000 ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((((fun (x1:(((eq Prop) (X Xx)) b)) (x2:(((eq Prop) b) (Y Xx)))=> (((eq_trans0000 x1) x2) (fun (x4:Prop)=> ((P (X Xx))->(P x4))))) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((((fun (x1:(((eq Prop) (X Xx)) b)) (x2:(((eq Prop) b) (Y Xx)))=> ((((eq_trans000 (Y Xx)) x1) x2) (fun (x4:Prop)=> ((P (X Xx))->(P x4))))) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (eq_trans000 (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((eq_trans00 (X Xx)) (Y Xx)) x1) x2) (fun (x4:Prop)=> ((P (X Xx))->(P x4))))) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) ((eq_trans00 (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> ((((((eq_trans0 (X Xx)) (X Xx)) (Y Xx)) x1) x2) (fun (x4:Prop)=> ((P (X Xx))->(P x4))))) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (((eq_trans0 (X Xx)) (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) x1) x2) (fun (x4:Prop)=> ((P (X Xx))->(P x4))))) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) ((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop))=> ((((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) x1) x2) (fun (x4:Prop)=> ((P (X Xx))->(P x4))))) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) ((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) (fun (x10:(P (X Xx)))=> x10))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x1:(P0 b)
% Instantiate: b:=(X Xx):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (X Xx))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (X Xx)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x1:(P (Y Xx))
% Instantiate: b:=(Y Xx):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_sym000:=(eq_sym00 b):((((eq Prop) (Y Xx)) b)->(((eq Prop) b) (Y Xx)))
% Instantiate: x:=(fun (x3:(fofType->Prop)) (x20:(fofType->Prop))=> (((eq Prop) (x3 Xx)) b)):((fofType->Prop)->((fofType->Prop)->Prop))
% Found (fun (x00:((x X) Y))=> eq_sym000) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> eq_sym000) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) b) (Y Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> eq_sym000)) as proof of (((eq Prop) b) (Y Xx))
% Found ((and_rect0 (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000)) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000)) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000)) as proof of (((eq Prop) b) (Y Xx))
% Found (eq_sym010 (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000))) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_sym01 (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000))) as proof of (((eq Prop) (Y Xx)) b)
% Found (((eq_sym0 b) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000))) as proof of (((eq Prop) (Y Xx)) b)
% Found (((eq_sym0 b) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000))) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_sym0000 (((eq_sym0 b) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000)))) x1) as proof of (P (Y Xx))
% Found ((eq_sym0000 (((eq_sym0 b) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000)))) x1) as proof of (P (Y Xx))
% Found (((fun (x2:(((eq Prop) (Y Xx)) b))=> ((eq_sym000 x2) P)) (((eq_sym0 b) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000)))) x1) as proof of (P (Y Xx))
% Found (((fun (x2:(((eq Prop) (Y Xx)) (X Xx)))=> (((eq_sym00 (X Xx)) x2) P)) (((eq_sym0 (X Xx)) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (eq_sym00 (X Xx)))))) x1) as proof of (P (Y Xx))
% Found (((fun (x2:(((eq Prop) (Y Xx)) (X Xx)))=> ((((eq_sym0 (Y Xx)) (X Xx)) x2) P)) (((eq_sym0 (X Xx)) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((eq_sym0 (Y Xx)) (X Xx)))))) x1) as proof of (P (Y Xx))
% Found (((fun (x2:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x2) P)) ((((eq_sym Prop) (X Xx)) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((eq_sym Prop) (Y Xx)) (X Xx)))))) x1) as proof of (P (Y Xx))
% Found (fun (x1:(P (X Xx)))=> (((fun (x2:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x2) P)) ((((eq_sym Prop) (X Xx)) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((eq_sym Prop) (Y Xx)) (X Xx)))))) x1)) as proof of (P (Y Xx))
% Found (fun (P:(Prop->Prop)) (x1:(P (X Xx)))=> (((fun (x2:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x2) P)) ((((eq_sym Prop) (X Xx)) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((eq_sym Prop) (Y Xx)) (X Xx)))))) x1)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found (((eq_trans000 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found ((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found (((((eq_trans0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_trans0000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found (((eq_trans000 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found ((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found (((((eq_trans0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found ((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_trans0000 ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found ((eq_trans000 (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((eq_trans00 (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found ((((eq_trans0 (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) (Y Xx)) (X Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found ((and_rect0 (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found ((eq_sym0000 (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((eq_sym0000 (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> ((eq_sym000 x1) (fun (x3:Prop)=> ((P (X Xx))->(P x3))))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((eq_sym00 (X Xx)) x1) (fun (x3:Prop)=> ((P (X Xx))->(P x3))))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> ((((eq_sym0 (Y Xx)) (X Xx)) x1) (fun (x3:Prop)=> ((P (X Xx))->(P x3))))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x1) (fun (x3:Prop)=> ((P (X Xx))->(P x3))))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop))=> (((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x1) (fun (x3:Prop)=> ((P (X Xx))->(P x3))))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop))=> (((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x1) (fun (x3:Prop)=> ((P (X Xx))->(P x3))))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) (fun (x10:(P (X Xx)))=> x10))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_trans0000 ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found ((eq_trans000 (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((eq_trans00 (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found ((((eq_trans0 (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))) as proof of (((x Y) X)->(((eq Prop) (Y Xx)) (X Xx)))
% Found (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) (Y Xx)) (X Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found ((and_rect0 (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))) as proof of (((eq Prop) (Y Xx)) (X Xx))
% Found (eq_sym0000 (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (eq_sym0000 (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> ((eq_sym000 x1) P)) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((eq_sym00 (X Xx)) x1) P)) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> ((((eq_sym0 (Y Xx)) (X Xx)) x1) P)) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x1) P)) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop))=> ((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x1) P)) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop))=> ((fun (x1:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x1) P)) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (Y Xx)) (X Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (Y Xx)) (Y Xx)) (X Xx)) ((eq_ref Prop) (Y Xx))))))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found x01:(P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P0 (f x))
% Found (fun (x01:(P0 (f x)))=> x01) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P0 (Y Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found x1:(P (X Xx))
% Instantiate: b:=(X Xx):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found x10:(P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P0 (X Xx))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found x10:(P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P0 (Y Xx))
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (((((eq_trans0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (eq_sym000 ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found ((eq_sym00 (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found (((eq_sym0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found ((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found ((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found x1:(P (X Xx))
% Instantiate: b:=(X Xx):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_trans000 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((((eq_trans0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_trans000 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((((eq_trans0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_trans000 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((((eq_trans0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_trans000 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((((eq_trans0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found x10:(P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P (X Xx))
% Found (fun (x10:(P (X Xx)))=> x10) as proof of (P0 (X Xx))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found eq_trans00010:=(eq_trans0001 (Y Xx)):((((eq Prop) b) (Y Xx))->(((eq Prop) (X Xx)) (Y Xx)))
% Found (eq_trans0001 (Y Xx)) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx)) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx)) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) b) (Y Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx)))) as proof of (((eq Prop) b) (Y Xx))
% Found ((and_rect0 (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx)))) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx)))) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx)))) as proof of (((eq Prop) b) (Y Xx))
% Found (((eq_trans00000 ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((eq_trans00000 ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((((fun (x1:(((eq Prop) (X Xx)) b)) (x2:(((eq Prop) b) (Y Xx)))=> (((eq_trans0000 x1) x2) (fun (x4:Prop)=> ((P (X Xx))->(P x4))))) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((((fun (x1:(((eq Prop) (X Xx)) b)) (x2:(((eq Prop) b) (Y Xx)))=> ((((eq_trans000 (Y Xx)) x1) x2) (fun (x4:Prop)=> ((P (X Xx))->(P x4))))) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((eq_trans000 c) x00)) (Y Xx))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((eq_trans00 (X Xx)) (Y Xx)) x1) x2) (fun (x4:Prop)=> ((P (X Xx))->(P x4))))) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> (((eq_trans00 (X Xx)) c) x00)) (Y Xx))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> ((((((eq_trans0 (X Xx)) (X Xx)) (Y Xx)) x1) x2) (fun (x4:Prop)=> ((P (X Xx))->(P x4))))) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> ((((eq_trans0 (X Xx)) (X Xx)) c) x00)) (Y Xx))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) x1) x2) (fun (x4:Prop)=> ((P (X Xx))->(P x4))))) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> (((((eq_trans Prop) (X Xx)) (X Xx)) c) x00)) (Y Xx))))) (fun (x10:(P (X Xx)))=> x10)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop))=> ((((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) x1) x2) (fun (x4:Prop)=> ((P (X Xx))->(P x4))))) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> (((((eq_trans Prop) (X Xx)) (X Xx)) c) x00)) (Y Xx))))) (fun (x10:(P (X Xx)))=> x10))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop))=> ((((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) x1) x2) (fun (x4:Prop)=> ((P (X Xx))->(P x4))))) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((fun (c:Prop)=> (((((eq_trans Prop) (X Xx)) (X Xx)) c) x00)) (Y Xx))))) (fun (x10:(P (X Xx)))=> x10))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found and_rect0000:=(and_rect000 (fun (x1:Prop)=> ((x Y) X))):(((x Y) X)->((x Y) X))
% Found (and_rect000 (fun (x1:Prop)=> ((x Y) X))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) b) (Y Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> ((and_rect00 eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found ((and_rect0 (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> (((and_rect0 (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X))))) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_trans00000 ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found ((eq_trans00000 ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (X Xx)) b)) (x2:(((eq Prop) b) (Y Xx)))=> (((eq_trans0000 x1) x2) P)) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) eq_trans0000) (fun (x1:Prop)=> ((x Y) X)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (X Xx)) b)) (x2:(((eq Prop) b) (Y Xx)))=> ((((eq_trans000 (Y Xx)) x1) x2) P)) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) b) (Y Xx))) (eq_trans000 (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((eq_trans00 (X Xx)) (Y Xx)) x1) x2) P)) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) ((eq_trans00 (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> ((((((eq_trans0 (X Xx)) (X Xx)) (Y Xx)) x1) x2) P)) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (((eq_trans0 (X Xx)) (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) x1) x2) P)) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) ((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X)))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop))=> (((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) x1) x2) P)) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) ((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X))))))) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop))=> (((fun (x1:(((eq Prop) (X Xx)) (X Xx))) (x2:(((eq Prop) (X Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) x1) x2) P)) ((eq_ref Prop) (X Xx))) (((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((fun (P1:Type) (x1:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x1) x0)) (((eq Prop) (X Xx)) (Y Xx))) ((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx))) (fun (x1:Prop)=> ((x Y) X))))))) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x1:(P0 b)
% Instantiate: b:=(X Xx):Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 (X Xx))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 (X Xx)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b)
% Found x1:(P (X Xx))
% Instantiate: a:=(X Xx):Prop
% Found x1 as proof of (P0 a)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found x1:(P (Y Xx))
% Instantiate: b:=(Y Xx):Prop
% Found x1 as proof of (P0 b)
% Found x1:(P (Y Xx))
% Instantiate: b:=(Y Xx):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_sym000:=(eq_sym00 b):((((eq Prop) (Y Xx)) b)->(((eq Prop) b) (Y Xx)))
% Instantiate: x:=(fun (x3:(fofType->Prop)) (x20:(fofType->Prop))=> (((eq Prop) (x3 Xx)) b)):((fofType->Prop)->((fofType->Prop)->Prop))
% Found (fun (x00:((x X) Y))=> eq_sym000) as proof of (((x Y) X)->(((eq Prop) b) (Y Xx)))
% Found (fun (x00:((x X) Y))=> eq_sym000) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) b) (Y Xx))))
% Found (and_rect00 (fun (x00:((x X) Y))=> eq_sym000)) as proof of (((eq Prop) b) (Y Xx))
% Found ((and_rect0 (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000)) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000)) as proof of (((eq Prop) b) (Y Xx))
% Found (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000)) as proof of (((eq Prop) b) (Y Xx))
% Found (eq_sym010 (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000))) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_sym01 (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000))) as proof of (((eq Prop) (Y Xx)) b)
% Found (((eq_sym0 b) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000))) as proof of (((eq Prop) (Y Xx)) b)
% Found (((eq_sym0 b) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000))) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_sym0000 (((eq_sym0 b) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000)))) x1) as proof of (P (Y Xx))
% Found ((eq_sym0000 (((eq_sym0 b) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000)))) x1) as proof of (P (Y Xx))
% Found (((fun (x2:(((eq Prop) (Y Xx)) b))=> ((eq_sym000 x2) P)) (((eq_sym0 b) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) b) (Y Xx))) (fun (x00:((x X) Y))=> eq_sym000)))) x1) as proof of (P (Y Xx))
% Found (((fun (x2:(((eq Prop) (Y Xx)) (X Xx)))=> (((eq_sym00 (X Xx)) x2) P)) (((eq_sym0 (X Xx)) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (eq_sym00 (X Xx)))))) x1) as proof of (P (Y Xx))
% Found (((fun (x2:(((eq Prop) (Y Xx)) (X Xx)))=> ((((eq_sym0 (Y Xx)) (X Xx)) x2) P)) (((eq_sym0 (X Xx)) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((eq_sym0 (Y Xx)) (X Xx)))))) x1) as proof of (P (Y Xx))
% Found (((fun (x2:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x2) P)) ((((eq_sym Prop) (X Xx)) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((eq_sym Prop) (Y Xx)) (X Xx)))))) x1) as proof of (P (Y Xx))
% Found (fun (x1:(P (X Xx)))=> (((fun (x2:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x2) P)) ((((eq_sym Prop) (X Xx)) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((eq_sym Prop) (Y Xx)) (X Xx)))))) x1)) as proof of (P (Y Xx))
% Found (fun (P:(Prop->Prop)) (x1:(P (X Xx)))=> (((fun (x2:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x2) P)) ((((eq_sym Prop) (X Xx)) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((eq_sym Prop) (Y Xx)) (X Xx)))))) x1)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop)) (x1:(P (X Xx)))=> (((fun (x2:(((eq Prop) (Y Xx)) (X Xx)))=> (((((eq_sym Prop) (Y Xx)) (X Xx)) x2) P)) ((((eq_sym Prop) (X Xx)) (Y Xx)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((eq_sym Prop) (Y Xx)) (X Xx)))))) x1)) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P0 (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x10:(P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P0 (Y Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (((((eq_trans0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (eq_sym0000 ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (eq_sym0000 ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> ((eq_sym000 x0) P0)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((eq_sym00 (f x)) x0) P0)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> ((((eq_sym0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) P0)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) P0)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) P0)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (((((eq_trans0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((eq_sym0000 ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((eq_sym0000 ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((eq_sym00 (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> ((((eq_sym0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found x1:(P (X Xx))
% Instantiate: a:=(X Xx):Prop
% Found x1 as proof of (P0 a)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found conj:(forall (A:Prop) (B:Prop), (A->(B->((and A) B))))
% Instantiate: b:=(forall (A:Prop) (B:Prop), (A->(B->((and A) B)))):Prop
% Found conj as proof of b
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (X Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X Xx))
% Found x10:(P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P0 (Y Xx))
% Found x10:(P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P0 (Y Xx))
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found x0:(P0 (f x))
% Instantiate: b:=(f x):Prop
% Found x0 as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x1:(P2 b)
% Instantiate: b:=(X Xx):Prop
% Found (fun (x1:(P2 b))=> x1) as proof of (P2 (X Xx))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 b))=> x1) as proof of ((P2 b)->(P2 (X Xx)))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 b))=> x1) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x1:(P2 b)
% Instantiate: b:=(X Xx):Prop
% Found (fun (x1:(P2 b))=> x1) as proof of (P2 (X Xx))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 b))=> x1) as proof of ((P2 b)->(P2 (X Xx)))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 b))=> x1) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P (f x))))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P (f x))))
% Found ((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P (f x))))
% Found (((((eq_trans0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P (f x))))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P (f x))))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P (f x))))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (eq_sym000 ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found ((eq_sym00 (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found (((eq_sym0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found ((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found ((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found eta_expansion000:=(eta_expansion00 b):(((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) (fun (x:((fofType->Prop)->((fofType->Prop)->Prop)))=> (b x)))
% Found (eta_expansion00 b) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) b0)
% Found ((eta_expansion0 Prop) b) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) b0)
% Found (((eta_expansion ((fofType->Prop)->((fofType->Prop)->Prop))) Prop) b) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) b0)
% Found (((eta_expansion ((fofType->Prop)->((fofType->Prop)->Prop))) Prop) b) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) b0)
% Found (((eta_expansion ((fofType->Prop)->((fofType->Prop)->Prop))) Prop) b) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) b) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) a)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) a)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) a)
% Found (eq_trans0010 ((eq_ref Prop) a)) as proof of (((x Y) X)->(((eq Prop) a) b))
% Found ((eq_trans001 b) ((eq_ref Prop) a)) as proof of (((x Y) X)->(((eq Prop) a) b))
% Found (((eq_trans00 a) b) ((eq_ref Prop) a)) as proof of (((x Y) X)->(((eq Prop) a) b))
% Found (((eq_trans00 a) b) ((eq_ref Prop) a)) as proof of (((x Y) X)->(((eq Prop) a) b))
% Found (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a))) as proof of (((x Y) X)->(((eq Prop) a) b))
% Found (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) a) b)))
% Found (and_rect00 (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a)))) as proof of (((eq Prop) a) b)
% Found ((and_rect0 (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a)))) as proof of (((eq Prop) a) b)
% Found (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a)))) as proof of (((eq Prop) a) b)
% Found (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a)))) as proof of (((eq Prop) a) b)
% Found (((eq_trans00000 (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a))))) ((eq_ref Prop) b)) x1) as proof of (P (Y Xx))
% Found (((eq_trans00000 (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a))))) ((eq_ref Prop) b)) x1) as proof of (P (Y Xx))
% Found ((((fun (x2:(((eq Prop) a) b)) (x3:(((eq Prop) b) (Y Xx)))=> (((eq_trans0000 x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a))))) ((eq_ref Prop) b)) x1) as proof of (P (Y Xx))
% Found ((((fun (x2:(((eq Prop) a) b)) (x3:(((eq Prop) b) (Y Xx)))=> ((((eq_trans000 (Y Xx)) x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a))))) ((eq_ref Prop) b)) x1) as proof of (P (Y Xx))
% Found ((((fun (x2:(((eq Prop) a) (Y Xx))) (x3:(((eq Prop) (Y Xx)) (Y Xx)))=> (((((eq_trans00 (Y Xx)) (Y Xx)) x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) (Y Xx))) (fun (x00:((x X) Y))=> (((eq_trans00 a) (Y Xx)) ((eq_ref Prop) a))))) ((eq_ref Prop) (Y Xx))) x1) as proof of (P (Y Xx))
% Found ((((fun (x2:(((eq Prop) (X Xx)) (Y Xx))) (x3:(((eq Prop) (Y Xx)) (Y Xx)))=> ((((((eq_trans0 (X Xx)) (Y Xx)) (Y Xx)) x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((eq_trans0 (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx)))))) ((eq_ref Prop) (Y Xx))) x1) as proof of (P (Y Xx))
% Found ((((fun (x2:(((eq Prop) (X Xx)) (Y Xx))) (x3:(((eq Prop) (Y Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (Y Xx)) (Y Xx)) x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx)))))) ((eq_ref Prop) (Y Xx))) x1) as proof of (P (Y Xx))
% Found (fun (x1:(P (X Xx)))=> ((((fun (x2:(((eq Prop) (X Xx)) (Y Xx))) (x3:(((eq Prop) (Y Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (Y Xx)) (Y Xx)) x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx)))))) ((eq_ref Prop) (Y Xx))) x1)) as proof of (P (Y Xx))
% Found (fun (P:(Prop->Prop)) (x1:(P (X Xx)))=> ((((fun (x2:(((eq Prop) (X Xx)) (Y Xx))) (x3:(((eq Prop) (Y Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (Y Xx)) (Y Xx)) x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx)))))) ((eq_ref Prop) (Y Xx))) x1)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found x1:(P b)
% Instantiate: b0:=b:Prop
% Found x1 as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_trans000 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((((eq_trans0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_trans000 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((((eq_trans0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) P0)) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found x1:(P1 (Y Xx))
% Instantiate: b:=(Y Xx):Prop
% Found x1 as proof of (P2 b)
% Found x1:(P1 (Y Xx))
% Instantiate: b:=(Y Xx):Prop
% Found x1 as proof of (P2 b)
% Found x1:(P (Y Xx))
% Instantiate: a:=(Y Xx):Prop
% Found x1 as proof of (P0 a)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found x1:(P b)
% Found x1 as proof of (P0 (X Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_trans000 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((((eq_trans0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found x02:(P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P0 (f x))
% Found (fun (x02:(P0 (f x)))=> x02) as proof of (P1 (f x))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((eq_trans00000 ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) b)) (x00:(((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_trans000 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) b)) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((((eq_trans0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (x00:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((((eq_trans Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x0) x00) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((eq_ref Prop) (f x))) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (x02:(P0 (f x)))=> x02))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found x20:(P1 (Y Xx))
% Found (fun (x20:(P1 (Y Xx)))=> x20) as proof of (P1 (Y Xx))
% Found (fun (x20:(P1 (Y Xx)))=> x20) as proof of (P2 (Y Xx))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))
% Found x10:(P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P2 b)
% Found x10:(P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found x1:(P0 b0)
% Instantiate: b0:=(X Xx):Prop
% Found (fun (x1:(P0 b0))=> x1) as proof of (P0 b)
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of ((P0 b0)->(P0 b))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found x1:(P0 b0)
% Instantiate: b0:=(X Xx):Prop
% Found (fun (x1:(P0 b0))=> x1) as proof of (P0 b)
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of ((P0 b0)->(P0 b))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of (P b0)
% Found x1:(P (X Xx))
% Instantiate: a:=(X Xx):Prop
% Found x1 as proof of (P0 a)
% Found x10:(P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P2 (Y Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found x10:(P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P2 (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))):(((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (eq_ref0 (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) b0)
% Found ((eq_ref (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) b0)
% Found ((eq_ref (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) b0)
% Found ((eq_ref (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) as proof of (((eq (((fofType->Prop)->((fofType->Prop)->Prop))->Prop)) (fun (R:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((and ((and (forall (Xx:(fofType->Prop)), ((R Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((R Xx) Xy)) ((R Xy) Xz))->((R Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((R X) Y)) ((R Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) b0)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b1)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b1)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b1)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (Y Xx))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (Y Xx))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (Y Xx))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (Y Xx))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:((fofType->Prop)->((fofType->Prop)->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 (f0 x)):(((eq Prop) (f0 x)) (f0 x))
% Found (eq_ref0 (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found ((eq_ref Prop) (f0 x)) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (((eq Prop) (f0 x)) (f x))
% Found (fun (x:((fofType->Prop)->((fofType->Prop)->Prop)))=> ((eq_ref Prop) (f0 x))) as proof of (forall (x:((fofType->Prop)->((fofType->Prop)->Prop))), (((eq Prop) (f0 x)) (f x)))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found x1:(P (Y Xx))
% Instantiate: b0:=(Y Xx):Prop
% Found x1 as proof of (P0 b0)
% Found x1:(P (Y Xx))
% Instantiate: b0:=(Y Xx):Prop
% Found x1 as proof of (P0 b0)
% Found x1:(P (Y Xx))
% Instantiate: b:=(Y Xx):Prop
% Found x1 as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found x1:(P (Y Xx))
% Found x1 as proof of (P0 (Y Xx))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (X Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (X Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found x10:(P b0)
% Found (fun (x10:(P b0))=> x10) as proof of (P b0)
% Found (fun (x10:(P b0))=> x10) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found x10:(P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P0 (Y Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P2 (Y Xx))
% Found x10:(P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P2 (Y Xx))
% Found x10:(P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P2 b)
% Found x10:(P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found x1:(P0 b0)
% Instantiate: b0:=(Y Xx):Prop
% Found (fun (x1:(P0 b0))=> x1) as proof of (P0 (Y Xx))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of ((P0 b0)->(P0 (Y Xx)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found x1:(P0 b)
% Instantiate: b0:=b:Prop
% Found (fun (x1:(P0 b))=> x1) as proof of (P0 b0)
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of ((P0 b)->(P0 b0))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b))=> x1) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (((((eq_trans0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((eq_sym0000 ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((eq_sym0000 ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((eq_sym00 (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> ((((eq_sym0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) (fun (x01:(P0 (f x)))=> x01))) as proof of (forall (P:(Prop->Prop)), ((P (f x))->(P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))))
% Found x10:(P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P0 (Y Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found (eq_sym010 ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_sym01 b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x))) as proof of (((eq Prop) b) (f x))
% Found ((eq_trans0000 ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P (f x))))
% Found (((eq_trans000 (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) b) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P (f x))))
% Found ((((eq_trans00 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P (f x))))
% Found (((((eq_trans0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P (f x))))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P (f x))))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (forall (P:(Prop->Prop)), ((P ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P (f x))))
% Found ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found (eq_sym0000 ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (eq_sym0000 ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> ((eq_sym000 x0) P0)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((eq_sym00 (f x)) x0) P0)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> ((((eq_sym0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) P0)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) (((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) P0)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) P0)) ((((((eq_trans Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))) ((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((eq_ref Prop) (f x)))))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found x1:(P b)
% Instantiate: b0:=b:Prop
% Found x1 as proof of (P0 b0)
% Found x1:(P (X Xx))
% Instantiate: a:=(X Xx):Prop
% Found x1 as proof of (P0 a)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (Y Xx))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found x10:(P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P0 (Y Xx))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found x10:(P b0)
% Found (fun (x10:(P b0))=> x10) as proof of (P b0)
% Found (fun (x10:(P b0))=> x10) as proof of (P0 b0)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b0)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found x10:(P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P0 (Y Xx))
% Found x10:(P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P (Y Xx))
% Found (fun (x10:(P (Y Xx)))=> x10) as proof of (P0 (Y Xx))
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found x10:(P b)
% Found (fun (x10:(P b))=> x10) as proof of (P b)
% Found (fun (x10:(P b))=> x10) as proof of (P0 b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x1:(P2 b)
% Instantiate: b:=(X Xx):Prop
% Found (fun (x1:(P2 b))=> x1) as proof of (P2 (X Xx))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 b))=> x1) as proof of ((P2 b)->(P2 (X Xx)))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 b))=> x1) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x1:(P2 b)
% Instantiate: b:=(X Xx):Prop
% Found (fun (x1:(P2 b))=> x1) as proof of (P2 (X Xx))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 b))=> x1) as proof of ((P2 b)->(P2 (X Xx)))
% Found (fun (P2:(Prop->Prop)) (x1:(P2 b))=> x1) as proof of (P1 b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) a)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) a)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) a)
% Found (eq_trans0010 ((eq_ref Prop) a)) as proof of (((x Y) X)->(((eq Prop) a) b))
% Found ((eq_trans001 b) ((eq_ref Prop) a)) as proof of (((x Y) X)->(((eq Prop) a) b))
% Found (((eq_trans00 a) b) ((eq_ref Prop) a)) as proof of (((x Y) X)->(((eq Prop) a) b))
% Found (((eq_trans00 a) b) ((eq_ref Prop) a)) as proof of (((x Y) X)->(((eq Prop) a) b))
% Found (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a))) as proof of (((x Y) X)->(((eq Prop) a) b))
% Found (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a))) as proof of (((x X) Y)->(((x Y) X)->(((eq Prop) a) b)))
% Found (and_rect00 (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a)))) as proof of (((eq Prop) a) b)
% Found ((and_rect0 (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a)))) as proof of (((eq Prop) a) b)
% Found (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a)))) as proof of (((eq Prop) a) b)
% Found (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a)))) as proof of (((eq Prop) a) b)
% Found (((eq_trans00000 (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a))))) ((eq_ref Prop) b)) x1) as proof of (P (Y Xx))
% Found (((eq_trans00000 (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a))))) ((eq_ref Prop) b)) x1) as proof of (P (Y Xx))
% Found ((((fun (x2:(((eq Prop) a) b)) (x3:(((eq Prop) b) (Y Xx)))=> (((eq_trans0000 x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a))))) ((eq_ref Prop) b)) x1) as proof of (P (Y Xx))
% Found ((((fun (x2:(((eq Prop) a) b)) (x3:(((eq Prop) b) (Y Xx)))=> ((((eq_trans000 (Y Xx)) x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) b)) (fun (x00:((x X) Y))=> (((eq_trans00 a) b) ((eq_ref Prop) a))))) ((eq_ref Prop) b)) x1) as proof of (P (Y Xx))
% Found ((((fun (x2:(((eq Prop) a) (Y Xx))) (x3:(((eq Prop) (Y Xx)) (Y Xx)))=> (((((eq_trans00 (Y Xx)) (Y Xx)) x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) a) (Y Xx))) (fun (x00:((x X) Y))=> (((eq_trans00 a) (Y Xx)) ((eq_ref Prop) a))))) ((eq_ref Prop) (Y Xx))) x1) as proof of (P (Y Xx))
% Found ((((fun (x2:(((eq Prop) (X Xx)) (Y Xx))) (x3:(((eq Prop) (Y Xx)) (Y Xx)))=> ((((((eq_trans0 (X Xx)) (Y Xx)) (Y Xx)) x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> ((((eq_trans0 (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx)))))) ((eq_ref Prop) (Y Xx))) x1) as proof of (P (Y Xx))
% Found ((((fun (x2:(((eq Prop) (X Xx)) (Y Xx))) (x3:(((eq Prop) (Y Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (Y Xx)) (Y Xx)) x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx)))))) ((eq_ref Prop) (Y Xx))) x1) as proof of (P (Y Xx))
% Found (fun (x1:(P (X Xx)))=> ((((fun (x2:(((eq Prop) (X Xx)) (Y Xx))) (x3:(((eq Prop) (Y Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (Y Xx)) (Y Xx)) x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx)))))) ((eq_ref Prop) (Y Xx))) x1)) as proof of (P (Y Xx))
% Found (fun (P:(Prop->Prop)) (x1:(P (X Xx)))=> ((((fun (x2:(((eq Prop) (X Xx)) (Y Xx))) (x3:(((eq Prop) (Y Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (Y Xx)) (Y Xx)) x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx)))))) ((eq_ref Prop) (Y Xx))) x1)) as proof of ((P (X Xx))->(P (Y Xx)))
% Found (fun (P:(Prop->Prop)) (x1:(P (X Xx)))=> ((((fun (x2:(((eq Prop) (X Xx)) (Y Xx))) (x3:(((eq Prop) (Y Xx)) (Y Xx)))=> (((((((eq_trans Prop) (X Xx)) (Y Xx)) (Y Xx)) x2) x3) P)) (((fun (P1:Type) (x2:(((x X) Y)->(((x Y) X)->P1)))=> (((((and_rect ((x X) Y)) ((x Y) X)) P1) x2) x0)) (((eq Prop) (X Xx)) (Y Xx))) (fun (x00:((x X) Y))=> (((((eq_trans Prop) (X Xx)) (X Xx)) (Y Xx)) ((eq_ref Prop) (X Xx)))))) ((eq_ref Prop) (Y Xx))) x1)) as proof of (forall (P:(Prop->Prop)), ((P (X Xx))->(P (Y Xx))))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (Y Xx))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (Y Xx))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (Y Xx))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (Y Xx))
% Found eq_ref00:=(eq_ref0 b00):(((eq Prop) b00) b00)
% Found (eq_ref0 b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found ((eq_ref Prop) b00) as proof of (((eq Prop) b00) b0)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b00)
% Found x1:(P b)
% Instantiate: b0:=b:Prop
% Found x1 as proof of (P0 b0)
% Found x1:(P b)
% Instantiate: b0:=b:Prop
% Found x1 as proof of (P0 b0)
% Found x0:(P0 (f x))
% Instantiate: a:=(f x):Prop
% Found x0 as proof of (P1 a)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found x0:(P0 (f x))
% Instantiate: a:=(f x):Prop
% Found x0 as proof of (P1 a)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found x1:(P1 (Y Xx))
% Instantiate: b:=(Y Xx):Prop
% Found x1 as proof of (P2 b)
% Found x1:(P1 (Y Xx))
% Instantiate: b:=(Y Xx):Prop
% Found x1 as proof of (P2 b)
% Found x1:(P1 (Y Xx))
% Instantiate: b:=(Y Xx):Prop
% Found x1 as proof of (P2 b)
% Found x1:(P1 (Y Xx))
% Instantiate: b:=(Y Xx):Prop
% Found x1 as proof of (P2 b)
% Found x1:(P (Y Xx))
% Instantiate: a:=(Y Xx):Prop
% Found x1 as proof of (P0 a)
% Found x1:(P (Y Xx))
% Instantiate: a:=(Y Xx):Prop
% Found x1 as proof of (P0 a)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found x1:(P b)
% Found x1 as proof of (P0 (X Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 a):(((eq Prop) a) a)
% Found (eq_ref0 a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found ((eq_ref Prop) a) as proof of (((eq Prop) a) b)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found x20:(P1 (Y Xx))
% Found (fun (x20:(P1 (Y Xx)))=> x20) as proof of (P1 (Y Xx))
% Found (fun (x20:(P1 (Y Xx)))=> x20) as proof of (P2 (Y Xx))
% Found x20:(P1 (Y Xx))
% Found (fun (x20:(P1 (Y Xx)))=> x20) as proof of (P1 (Y Xx))
% Found (fun (x20:(P1 (Y Xx)))=> x20) as proof of (P2 (Y Xx))
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b0)
% Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% Found (eq_ref0 b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (Y Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) b)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b)
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x10:(P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P2 b)
% Found x10:(P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P2 b)
% Found x10:(P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found x1:(P0 b0)
% Instantiate: b0:=(X Xx):Prop
% Found (fun (x1:(P0 b0))=> x1) as proof of (P0 b)
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of ((P0 b0)->(P0 b))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of (P b0)
% Found x10:(P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P1 b)
% Found (fun (x10:(P1 b))=> x10) as proof of (P2 b)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found x1:(P0 b0)
% Instantiate: b0:=(X Xx):Prop
% Found (fun (x1:(P0 b0))=> x1) as proof of (P0 b)
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of ((P0 b0)->(P0 b))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found x1:(P0 b0)
% Instantiate: b0:=(X Xx):Prop
% Found (fun (x1:(P0 b0))=> x1) as proof of (P0 b)
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of ((P0 b0)->(P0 b))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of (P b0)
% Found iff_sym:=(fun (A:Prop) (B:Prop) (H:((iff A) B))=> ((((conj (B->A)) (A->B)) (((proj2 (A->B)) (B->A)) H)) (((proj1 (A->B)) (B->A)) H))):(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A)))
% Instantiate: a:=(forall (A:Prop) (B:Prop), (((iff A) B)->((iff B) A))):Prop
% Found iff_sym as proof of a
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found x10:(P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P2 (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x10:(P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P2 (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x10:(P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P2 (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found x10:(P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P1 (Y Xx))
% Found (fun (x10:(P1 (Y Xx)))=> x10) as proof of (P2 (Y Xx))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (X Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b)
% Found eq_ref00:=(eq_ref0 (X Xx)):(((eq Prop) (X Xx)) (X Xx))
% Found (eq_ref0 (X Xx)) as proof of (((eq Prop) (X Xx)) b1)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b1)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b1)
% Found ((eq_ref Prop) (X Xx)) as proof of (((eq Prop) (X Xx)) b1)
% Found eq_ref00:=(eq_ref0 b1):(((eq Prop) b1) b1)
% Found (eq_ref0 b1) as proof of (((eq Prop) b1) (Y Xx))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (Y Xx))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (Y Xx))
% Found ((eq_ref Prop) b1) as proof of (((eq Prop) b1) (Y Xx))
% Found eq_ref00:=(eq_ref0 (Y Xx)):(((eq Prop) (Y Xx)) (Y Xx))
% Found (eq_ref0 (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found ((eq_ref Prop) (Y Xx)) as proof of (((eq Prop) (Y Xx)) b0)
% Found x1:(P0 b0)
% Instantiate: b0:=(X Xx):Prop
% Found (fun (x1:(P0 b0))=> x1) as proof of (P0 (X Xx))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of ((P0 b0)->(P0 (X Xx)))
% Found (fun (P0:(Prop->Prop)) (x1:(P0 b0))=> x1) as proof of (P b0)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found eq_ref00:=(eq_ref0 a):(((eq fofType) a) a)
% Found (eq_ref0 a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found ((eq_ref fofType) a) as proof of (((eq fofType) a) Xx)
% Found x1:(P (Y Xx))
% Instantiate: b0:=(Y Xx):Prop
% Found x1 as proof of (P0 b0)
% Found x1:(P (Y Xx))
% Instantiate: b0:=(Y Xx):Prop
% Found x1 as proof of (P0 b0)
% Found x1:(P (Y Xx))
% Instantiate: b0:=(Y Xx):Prop
% Found x1 as proof of (P0 b0)
% Found x1:(P (Y Xx))
% Instantiate: b0:=(Y Xx):Prop
% Found x1 as proof of (P0 b0)
% Found x1:(P (Y Xx))
% Instantiate: b0:=(Y Xx):Prop
% Found x1 as proof of (P0 b0)
% Found x1:(P b)
% Instantiate: b0:=b:Prop
% Found x1 as proof of (P0 b0)
% Found x0:(P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Instantiate: b:=((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))):Prop
% Found x0 as proof of (P3 b)
% Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) b)
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P2 (f x))
% Found ((eq_sym0100 ((eq_ref Prop) (f x))) x0) as proof of (P2 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) b))=> ((eq_sym010 x00) P2)) ((eq_ref Prop) (f x))) x0) as proof of (P2 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((eq_sym01 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x00) P2)) ((eq_ref Prop) (f x))) x0) as proof of (P2 (f x))
% Found (((fun (x00:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x00) P2)) ((eq_ref Prop) (f x))) x0) as proof of (P2 (f x))
% Found (fun (x0:(P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x00) P2)) ((eq_ref Prop) (f x))) x0)) as proof of (P2 (f x))
% Found (fun (P2:(Prop->Prop)) (x0:(P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x00) P2)) ((eq_ref Prop) (f x))) x0)) as proof of ((P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P2 (f x)))
% Found (fun (P2:(Prop->Prop)) (x0:(P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x00) P2)) ((eq_ref Prop) (f x))) x0)) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x))
% Found ((eq_sym0000 (fun (P2:(Prop->Prop)) (x0:(P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x00) P2)) ((eq_ref Prop) (f x))) x0))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found ((eq_sym0000 (fun (P2:(Prop->Prop)) (x0:(P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x00) P2)) ((eq_ref Prop) (f x))) x0))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x00) P2)) ((eq_ref Prop) (f x))) x0))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((eq_sym00 (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x00) P2)) ((eq_ref Prop) (f x))) x0))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> ((((eq_sym0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> ((((eq_sym0 (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x00) P2)) ((eq_ref Prop) (f x))) x0))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x00) P2)) ((eq_ref Prop) (f x))) x0))) (fun (x01:(P0 (f x)))=> x01)) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)))=> (((((eq_sym Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) (f x)) x0) (fun (x1:Prop)=> ((P0 (f x))->(P0 x1))))) (fun (P2:(Prop->Prop)) (x0:(P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((fun (x00:(((eq Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))=> (((((eq_sym Prop) (f x)) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) x00) P2)) ((eq_ref Prop) (f x))) x0))) (fun (x01:(P0 (f x)))=> x01))) as proof of ((P0 (f x))->(P0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref000:=(eq_ref00 P2):((P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (eq_ref00 P2) as proof of (P3 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) P2) as proof of (P3 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) P2) as proof of (P3 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) P2) as proof of (P3 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found eq_ref000:=(eq_ref00 P2):((P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))->(P2 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))))
% Found (eq_ref00 P2) as proof of (P3 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found ((eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) P2) as proof of (P3 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) P2) as proof of (P3 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) P2) as proof of (P3 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (f x))
% Found eq_ref00:=(eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))):(((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx)))))))
% Found (eq_ref0 ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found ((eq_ref Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) as proof of (((eq Prop) ((and ((and (forall (Xx:(fofType->Prop)), ((x Xx) Xx))) (forall (Xx:(fofType->Prop)) (Xy:(fofType->Prop)) (Xz:(fofType->Prop)), (((and ((x Xx) Xy)) ((x Xy) Xz))->((x Xx) Xz))))) (forall (X:(fofType->Prop)) (Y:(fofType->Prop)), (((and ((x X) Y)) ((x Y) X))->(forall (Xx:fofType), (((eq Prop) (X Xx)) (Y Xx))))))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% Found (eq_ref0 b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b0)
% Found ((eq_ref Prop) b) as proof of (((eq Prop) b) b
% EOF
%------------------------------------------------------------------------------