TSTP Solution File: SYO206^5 by cocATP---0.2.0

%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SYO206^5 : TPTP v7.5.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n005.cluster.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory   : 8042.1875MB
% OS       : Linux 3.10.0-693.el7.x86_64
% CPULimit : 300s
% WCLimit  : 0s
% DateTime : Tue Mar 29 00:50:54 EDT 2022

% Result   : Timeout 300.04s 300.23s
% Output   : None 
% Verified : 
% SZS Type : -

% Comments : 
%------------------------------------------------------------------------------
%----No solution output by system
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.11  % Problem    : SYO206^5 : TPTP v7.5.0. Released v4.0.0.
% 0.13/0.12  % Command    : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% 0.13/0.33  % Computer   : n005.cluster.edu
% 0.13/0.33  % Model      : x86_64 x86_64
% 0.13/0.33  % CPUModel   : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.13/0.33  % RAMPerCPU  : 8042.1875MB
% 0.13/0.33  % OS         : Linux 3.10.0-693.el7.x86_64
% 0.13/0.33  % CPULimit   : 300
% 0.13/0.33  % DateTime   : Fri Mar 11 18:28:12 EST 2022
% 0.13/0.33  % CPUTime    : 
% 0.20/0.34  ModuleCmd_Load.c(213):ERROR:105: Unable to locate a modulefile for 'python/python27'
% 0.20/0.34  Python 2.7.5
% 0.86/1.02  Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% 0.86/1.02  FOF formula ((ex Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) of role conjecture named cCT265
% 0.86/1.02  Conjecture to prove = ((ex Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))):Prop
% 0.86/1.02  We need to prove ['((ex Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))']
% 0.86/1.02  Trying to prove ((ex Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 0.86/1.02  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 0.86/1.02  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 0.86/1.02  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 0.86/1.02  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 0.86/1.02  Found eq_ref000:=(eq_ref00 P):((P (((eq Prop) x) Xy))->(P (((eq Prop) x) Xy)))
% 0.86/1.02  Found (eq_ref00 P) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 0.86/1.02  Found ((eq_ref0 (((eq Prop) x) Xy)) P) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 0.86/1.02  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 0.86/1.02  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 0.86/1.02  Found (fun (P:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 0.86/1.02  Found x0:(P (((eq Prop) x) Xy))
% 0.86/1.02  Instantiate: x:=Xy:Prop
% 0.86/1.02  Found (fun (x0:(P (((eq Prop) x) Xy)))=> x0) as proof of (P (((eq Prop) Xy) x))
% 0.86/1.02  Found (fun (P:(Prop->Prop)) (x0:(P (((eq Prop) x) Xy)))=> x0) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 0.86/1.02  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 0.86/1.02  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 0.86/1.02  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 0.86/1.02  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 0.86/1.02  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 0.86/1.02  Found (eq_sym000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 0.86/1.02  Found ((eq_sym00 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 0.86/1.02  Found (((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 0.86/1.02  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 0.86/1.02  Found eq_ref00:=(eq_ref0 (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))):(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 0.86/1.02  Found (eq_ref0 (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 0.86/1.02  Found ((eq_ref (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 0.86/1.02  Found ((eq_ref (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 0.86/1.02  Found ((eq_ref (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 1.18/1.34  Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% 1.18/1.34  Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))))
% 1.18/1.34  Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))))
% 1.18/1.34  Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))))
% 1.18/1.34  Found (fun (x:Prop)=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))))
% 1.18/1.34  Found (fun (x:Prop)=> ((eq_ref Prop) (f x))) as proof of (forall (x:Prop), (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 1.18/1.34  Found eq_ref00:=(eq_ref0 (f x)):(((eq Prop) (f x)) (f x))
% 1.18/1.34  Found (eq_ref0 (f x)) as proof of (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))))
% 1.18/1.34  Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))))
% 1.18/1.34  Found ((eq_ref Prop) (f x)) as proof of (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))))
% 1.18/1.34  Found (fun (x:Prop)=> ((eq_ref Prop) (f x))) as proof of (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))))
% 1.18/1.34  Found (fun (x:Prop)=> ((eq_ref Prop) (f x))) as proof of (forall (x:Prop), (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 1.18/1.34  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 1.18/1.34  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.18/1.34  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.18/1.34  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.18/1.34  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.18/1.34  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.18/1.34  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.18/1.34  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) P)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.18/1.34  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) P)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.18/1.34  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.18/1.34  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.18/1.34  Found (fun (P:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.18/1.34  Found eq_ref000:=(eq_ref00 P):((P (((eq Prop) x) Xy))->(P (((eq Prop) x) Xy)))
% 1.18/1.34  Found (eq_ref00 P) as proof of (P0 (((eq Prop) x) Xy))
% 1.18/1.34  Found ((eq_ref0 (((eq Prop) x) Xy)) P) as proof of (P0 (((eq Prop) x) Xy))
% 1.18/1.34  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P) as proof of (P0 (((eq Prop) x) Xy))
% 1.18/1.34  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P) as proof of (P0 (((eq Prop) x) Xy))
% 1.18/1.34  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 1.18/1.34  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.33/1.48  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.33/1.48  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.33/1.48  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.33/1.48  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.33/1.48  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.33/1.48  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.33/1.48  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.33/1.48  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.33/1.48  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.33/1.48  Found (fun (P:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.33/1.48  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 1.33/1.48  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 1.33/1.48  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 1.33/1.48  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 1.33/1.48  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 1.33/1.48  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 1.33/1.48  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.33/1.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.33/1.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.33/1.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.33/1.48  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 1.33/1.48  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.33/1.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.33/1.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.33/1.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.33/1.48  Found ((eq_trans0000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 1.33/1.48  Found (((eq_trans000 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 1.33/1.48  Found ((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 1.33/1.48  Found (((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 1.56/1.73  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 1.56/1.73  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 1.56/1.73  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.56/1.73  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.56/1.73  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.56/1.73  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.56/1.73  Found eq_ref000:=(eq_ref00 P):((P (((eq Prop) x) Xy))->(P (((eq Prop) x) Xy)))
% 1.56/1.73  Found (eq_ref00 P) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.56/1.73  Found ((eq_ref0 (((eq Prop) x) Xy)) P) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.56/1.73  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.56/1.73  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.56/1.73  Found (fun (P:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.56/1.73  Found (fun (P:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.56/1.73  Found x0:(P (((eq Prop) x) Xy))
% 1.56/1.73  Instantiate: x:=Xy:Prop
% 1.56/1.73  Found (fun (x0:(P (((eq Prop) x) Xy)))=> x0) as proof of (P (((eq Prop) Xy) x))
% 1.56/1.73  Found (fun (P:(Prop->Prop)) (x0:(P (((eq Prop) x) Xy)))=> x0) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.56/1.73  Found (fun (P:(Prop->Prop)) (x0:(P (((eq Prop) x) Xy)))=> x0) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.56/1.73  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 1.56/1.73  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.56/1.73  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.56/1.73  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.56/1.73  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.56/1.73  Found (eq_sym000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.56/1.73  Found ((eq_sym00 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.56/1.73  Found (((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.56/1.73  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.56/1.73  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.56/1.73  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 1.56/1.73  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.56/1.73  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.56/1.73  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.56/1.73  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.56/1.73  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.76/1.95  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.76/1.95  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) P)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.76/1.95  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) P)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.76/1.95  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.76/1.95  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.76/1.95  Found (fun (P:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.76/1.95  Found (fun (P:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.76/1.95  Found eq_ref000:=(eq_ref00 P):((P (((eq Prop) x) Xy))->(P (((eq Prop) x) Xy)))
% 1.76/1.95  Found (eq_ref00 P) as proof of (P0 (((eq Prop) x) Xy))
% 1.76/1.95  Found ((eq_ref0 (((eq Prop) x) Xy)) P) as proof of (P0 (((eq Prop) x) Xy))
% 1.76/1.95  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P) as proof of (P0 (((eq Prop) x) Xy))
% 1.76/1.95  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P) as proof of (P0 (((eq Prop) x) Xy))
% 1.76/1.95  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 1.76/1.95  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.76/1.95  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.76/1.95  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.76/1.95  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 1.76/1.95  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.76/1.95  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.76/1.95  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.76/1.95  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.76/1.95  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.76/1.95  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.76/1.95  Found (fun (P:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 1.97/2.19  Found (fun (P:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.97/2.19  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 1.97/2.19  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 1.97/2.19  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 1.97/2.19  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 1.97/2.19  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 1.97/2.19  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 1.97/2.19  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.97/2.19  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.97/2.19  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.97/2.19  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.97/2.19  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 1.97/2.19  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.97/2.19  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.97/2.19  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.97/2.19  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 1.97/2.19  Found ((eq_trans0000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.97/2.19  Found (((eq_trans000 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.97/2.19  Found ((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.97/2.19  Found (((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.97/2.19  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.97/2.19  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 1.97/2.19  Found eq_ref00:=(eq_ref0 a):(((eq (Prop->Prop)) a) a)
% 1.97/2.19  Found (eq_ref0 a) as proof of (((eq (Prop->Prop)) a) b)
% 1.97/2.19  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b)
% 1.97/2.19  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b)
% 1.97/2.19  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b)
% 1.97/2.19  Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (Prop->Prop)) b) (fun (x:Prop)=> (b x)))
% 1.97/2.19  Found (eta_expansion_dep00 b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 1.97/2.19  Found ((eta_expansion_dep0 (fun (x1:Prop)=> Prop)) b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 1.97/2.19  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 2.17/2.38  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 2.17/2.38  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 2.17/2.38  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 2.17/2.38  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 2.17/2.38  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 2.17/2.38  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 2.17/2.38  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 2.17/2.38  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 2.17/2.38  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.17/2.38  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.17/2.38  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.17/2.38  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.17/2.38  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 2.17/2.38  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.17/2.38  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.17/2.38  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.17/2.38  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.17/2.38  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.17/2.38  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.17/2.38  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) P)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.17/2.38  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) P)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.17/2.38  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) P)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.17/2.38  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) P)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.17/2.38  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) P)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.17/2.38  Found (fun (P:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) P)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.46/2.64  Found eq_ref000:=(eq_ref00 P):((P (((eq Prop) x) Xy))->(P (((eq Prop) x) Xy)))
% 2.46/2.64  Found (eq_ref00 P) as proof of (P0 (((eq Prop) x) Xy))
% 2.46/2.64  Found ((eq_ref0 (((eq Prop) x) Xy)) P) as proof of (P0 (((eq Prop) x) Xy))
% 2.46/2.64  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P) as proof of (P0 (((eq Prop) x) Xy))
% 2.46/2.64  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P) as proof of (P0 (((eq Prop) x) Xy))
% 2.46/2.64  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 2.46/2.64  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 2.46/2.64  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 2.46/2.64  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 2.46/2.64  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 2.46/2.64  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 2.46/2.64  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.46/2.64  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.46/2.64  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.46/2.64  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.46/2.64  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 2.46/2.64  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.46/2.64  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.46/2.64  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.46/2.64  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.46/2.64  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.46/2.64  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.46/2.64  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.46/2.64  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.46/2.64  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.46/2.64  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.46/2.64  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.78/2.99  Found (fun (P:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.78/2.99  Found x0:(P0 (((eq Prop) x) Xy))
% 2.78/2.99  Instantiate: x:=Xy:Prop
% 2.78/2.99  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 2.78/2.99  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 2.78/2.99  Found x0:(P0 (((eq Prop) x) Xy))
% 2.78/2.99  Instantiate: x:=Xy:Prop
% 2.78/2.99  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 2.78/2.99  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 2.78/2.99  Found x0:(P0 (((eq Prop) x) Xy))
% 2.78/2.99  Instantiate: x:=Xy:Prop
% 2.78/2.99  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 2.78/2.99  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 2.78/2.99  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 2.78/2.99  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 2.78/2.99  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 2.78/2.99  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 2.78/2.99  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 2.78/2.99  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 2.78/2.99  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.78/2.99  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.78/2.99  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.78/2.99  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.78/2.99  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 2.78/2.99  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.78/2.99  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.78/2.99  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.78/2.99  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 2.78/2.99  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.78/2.99  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.78/2.99  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) P)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.78/2.99  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) P)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.78/2.99  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) P)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 2.78/2.99  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) P)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 3.07/3.22  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) P)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 3.07/3.22  Found (fun (P:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) P)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 3.07/3.22  Found (fun (P:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) P)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.07/3.22  Found eq_ref000:=(eq_ref00 P):((P (((eq Prop) x) Xy))->(P (((eq Prop) x) Xy)))
% 3.07/3.22  Found (eq_ref00 P) as proof of (P0 (((eq Prop) x) Xy))
% 3.07/3.22  Found ((eq_ref0 (((eq Prop) x) Xy)) P) as proof of (P0 (((eq Prop) x) Xy))
% 3.07/3.22  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P) as proof of (P0 (((eq Prop) x) Xy))
% 3.07/3.22  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P) as proof of (P0 (((eq Prop) x) Xy))
% 3.07/3.22  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 3.07/3.22  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 3.07/3.22  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 3.07/3.22  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 3.07/3.22  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 3.07/3.22  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 3.07/3.22  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 3.07/3.22  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 3.07/3.22  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 3.07/3.22  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 3.07/3.22  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 3.07/3.22  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 3.07/3.22  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 3.07/3.22  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 3.07/3.22  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 3.07/3.22  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 3.07/3.22  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 3.07/3.22  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 3.07/3.22  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 3.07/3.22  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 3.23/3.38  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 3.23/3.38  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P)) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 3.23/3.38  Found (fun (P:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P))) as proof of ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x)))
% 3.23/3.38  Found (fun (P:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (x00:(((eq Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P (((eq Prop) x) Xy))->(P x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P1:(Prop->Prop)), ((P1 x)->(P1 Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.23/3.38  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 3.23/3.38  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.23/3.38  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.23/3.38  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.23/3.38  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.23/3.38  Found x0:(P0 (((eq Prop) x) Xy))
% 3.23/3.38  Instantiate: x:=Xy:Prop
% 3.23/3.38  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 3.23/3.38  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.23/3.38  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.23/3.38  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 3.23/3.38  Found (eq_ref00 P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.23/3.38  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.30/3.48  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 3.30/3.48  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.30/3.48  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.30/3.48  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.30/3.48  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.30/3.48  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 3.30/3.48  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.30/3.48  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.30/3.48  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.30/3.48  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.30/3.48  Found x0:(P0 (((eq Prop) x) Xy))
% 3.30/3.48  Instantiate: x:=Xy:Prop
% 3.30/3.48  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 3.30/3.48  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.30/3.48  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 3.30/3.48  Found (eq_ref00 P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.30/3.48  Found x0:(P0 (((eq Prop) x) Xy))
% 3.30/3.48  Instantiate: x:=Xy:Prop
% 3.30/3.48  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 3.30/3.48  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.30/3.48  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 3.30/3.48  Found (eq_ref00 P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.30/3.48  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 3.85/3.99  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 3.85/3.99  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 3.85/3.99  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 3.85/3.99  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 3.85/3.99  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 3.85/3.99  Found (eq_ref00 P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found x0:(P0 (((eq Prop) x) Xy))
% 3.85/3.99  Instantiate: x:=Xy:Prop
% 3.85/3.99  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 3.85/3.99  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 3.85/3.99  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 3.85/3.99  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 3.85/3.99  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 3.85/3.99  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 3.85/3.99  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 3.85/3.99  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 3.85/3.99  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 3.85/3.99  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 3.85/3.99  Found (eq_ref00 P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found x0:(P0 (((eq Prop) x) Xy))
% 3.85/3.99  Instantiate: x:=Xy:Prop
% 3.85/3.99  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 3.85/3.99  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 3.85/3.99  Found (eq_ref00 P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found x0:(P0 (((eq Prop) x) Xy))
% 3.85/3.99  Instantiate: x:=Xy:Prop
% 3.85/3.99  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 3.85/3.99  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 3.85/3.99  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 3.85/3.99  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.03/4.22  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.03/4.22  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.03/4.22  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.03/4.22  Found (eq_sym100 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found ((eq_sym10 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found (((eq_sym1 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 4.03/4.22  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.03/4.22  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.03/4.22  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.03/4.22  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.03/4.22  Found (eq_sym000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found ((eq_sym00 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found (((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 4.03/4.22  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.03/4.22  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.03/4.22  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.03/4.22  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.03/4.22  Found (eq_sym000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found ((eq_sym00 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found (((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.03/4.22  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 4.03/4.22  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.43/4.58  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.43/4.58  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.43/4.58  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.43/4.58  Found (eq_sym100 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 4.43/4.58  Found ((eq_sym10 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 4.43/4.58  Found (((eq_sym1 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 4.43/4.58  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 4.43/4.58  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 4.43/4.58  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.43/4.58  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.43/4.58  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.43/4.58  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.43/4.58  Found (eq_sym000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 4.43/4.58  Found ((eq_sym00 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 4.43/4.58  Found (((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 4.43/4.58  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 4.43/4.58  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 4.43/4.58  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.43/4.58  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.43/4.58  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.43/4.58  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.43/4.58  Found (eq_sym000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 4.43/4.58  Found ((eq_sym00 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 4.43/4.58  Found (((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 4.43/4.58  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 4.43/4.58  Found x0:(P0 (((eq Prop) x) Xy))
% 4.43/4.58  Instantiate: x:=Xy:Prop
% 4.43/4.58  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 4.43/4.58  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.43/4.58  Found eta_expansion000:=(eta_expansion00 (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))):(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) (fun (x:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 4.43/4.58  Found (eta_expansion00 (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 4.43/4.58  Found ((eta_expansion0 Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 4.49/4.65  Found (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 4.49/4.65  Found (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 4.49/4.65  Found (eq_sym0001 (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.49/4.65  Found (eq_sym0001 (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.49/4.65  Found ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.49/4.65  Found ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.49/4.65  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.49/4.65  Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 4.49/4.65  Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 4.49/4.65  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 4.49/4.65  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 4.49/4.65  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:Prop), (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 4.49/4.65  Found (functional_extensionality00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.49/4.65  Found (functional_extensionality00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.49/4.65  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.49/4.65  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.49/4.65  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.64/4.83  Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 4.64/4.83  Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 4.64/4.83  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 4.64/4.83  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 4.64/4.83  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:Prop), (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 4.64/4.83  Found (functional_extensionality_dep00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.64/4.83  Found (functional_extensionality_dep00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.64/4.83  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.64/4.83  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.64/4.83  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.64/4.83  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 4.64/4.83  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.64/4.83  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.64/4.83  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.64/4.83  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.64/4.83  Found x0:(P0 (((eq Prop) x) Xy))
% 4.64/4.83  Instantiate: x:=Xy:Prop
% 4.64/4.83  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 4.64/4.83  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.64/4.83  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.64/4.83  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 4.64/4.83  Found (eq_ref00 P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.64/4.83  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.64/4.83  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.64/4.83  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.64/4.83  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.64/4.83  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.64/4.83  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 4.64/4.83  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.64/4.83  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.88/5.08  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.88/5.08  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.88/5.08  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 4.88/5.08  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.88/5.08  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.88/5.08  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.88/5.08  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.88/5.08  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 4.88/5.08  Found (eq_ref00 P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.88/5.08  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.88/5.08  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.88/5.08  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.88/5.08  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.88/5.08  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.88/5.08  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 4.88/5.08  Found (eq_ref00 P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.88/5.08  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.88/5.08  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.88/5.08  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.88/5.08  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.88/5.08  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.88/5.08  Found x0:(P0 (((eq Prop) x) Xy))
% 4.88/5.08  Instantiate: x:=Xy:Prop
% 4.88/5.08  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 4.88/5.08  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.88/5.08  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.88/5.08  Found x0:(P0 (((eq Prop) x) Xy))
% 4.88/5.08  Instantiate: x:=Xy:Prop
% 4.88/5.08  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 4.88/5.08  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 4.88/5.08  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.88/5.08  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 4.88/5.08  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.88/5.08  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.88/5.08  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.88/5.08  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 4.88/5.08  Found (eq_sym100 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 4.88/5.08  Found ((eq_sym10 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found (((eq_sym1 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 5.16/5.35  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.16/5.35  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.16/5.35  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.16/5.35  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.16/5.35  Found (eq_sym000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found ((eq_sym00 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found (((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 5.16/5.35  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.16/5.35  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.16/5.35  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.16/5.35  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.16/5.35  Found (eq_sym000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found ((eq_sym00 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found (((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 5.16/5.35  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.16/5.35  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 5.16/5.35  Found (eq_ref00 P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 5.75/5.89  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 5.75/5.89  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 5.75/5.89  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 5.75/5.89  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 5.75/5.89  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.75/5.89  Found x0:(P0 (((eq Prop) x) Xy))
% 5.75/5.89  Instantiate: x:=Xy:Prop
% 5.75/5.89  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 5.75/5.89  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 5.75/5.89  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.75/5.89  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 5.75/5.89  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 5.75/5.89  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 5.75/5.89  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 5.75/5.89  Found x0:(P0 (((eq Prop) x) Xy))
% 5.75/5.89  Instantiate: x:=Xy:Prop
% 5.75/5.89  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 5.75/5.89  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 5.75/5.89  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 5.75/5.89  Found (eq_ref00 P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 5.75/5.89  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 5.75/5.89  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 5.75/5.89  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 5.75/5.89  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 5.75/5.89  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 5.75/5.89  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.75/5.89  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.75/5.89  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.75/5.89  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.75/5.89  Found (eq_sym000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.75/5.89  Found ((eq_sym00 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.75/5.89  Found (((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.75/5.89  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.75/5.89  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 5.75/5.89  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 5.75/5.89  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.75/5.89  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 5.75/5.89  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.08/6.29  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.08/6.29  Found (eq_sym1000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.08/6.29  Found (eq_sym1000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.08/6.29  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym100 x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.08/6.29  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym10 (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.08/6.29  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym1 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.08/6.29  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.08/6.29  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.08/6.29  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 6.08/6.29  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 6.08/6.29  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 6.08/6.29  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 6.08/6.29  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 6.08/6.29  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 6.08/6.29  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 6.08/6.29  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.08/6.29  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.08/6.29  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.08/6.29  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.08/6.29  Found ((eq_sym1000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.08/6.29  Found ((eq_sym1000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.08/6.29  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym100 x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.08/6.29  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym10 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.08/6.29  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym1 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.08/6.29  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.25/6.47  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.25/6.47  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 6.25/6.47  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 6.25/6.47  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 6.25/6.47  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 6.25/6.47  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 6.25/6.47  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 6.25/6.47  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 6.25/6.47  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 6.25/6.47  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 6.25/6.47  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 6.25/6.47  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 6.25/6.47  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 6.25/6.47  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 6.25/6.47  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 6.25/6.47  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 6.25/6.47  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 6.25/6.47  Found ((eq_trans0000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 6.25/6.47  Found (((eq_trans000 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 6.25/6.47  Found ((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 6.25/6.47  Found (((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 6.25/6.47  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 6.25/6.47  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 6.25/6.47  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 6.25/6.47  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.25/6.47  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.25/6.47  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.25/6.47  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.25/6.47  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.54  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.54  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.54  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.54  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.54  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.54  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.54  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 6.33/6.54  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 6.33/6.54  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.33/6.54  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.33/6.54  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.33/6.54  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.33/6.55  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.55  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.55  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.55  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.55  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.55  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.55  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.33/6.55  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 6.33/6.55  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 6.33/6.55  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.33/6.55  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.33/6.55  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.58/6.76  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.58/6.76  Found (eq_sym000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 6.58/6.76  Found ((eq_sym00 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 6.58/6.76  Found (((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 6.58/6.76  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 6.58/6.76  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 6.58/6.76  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 6.58/6.76  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 6.58/6.76  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 6.58/6.76  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 6.58/6.76  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 6.58/6.76  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.58/6.76  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.58/6.76  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.58/6.76  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.58/6.76  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.58/6.76  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.58/6.76  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.58/6.76  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.58/6.76  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.58/6.76  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.58/6.76  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.58/6.76  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 6.58/6.76  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 6.58/6.76  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 6.58/6.76  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 6.73/6.93  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 6.73/6.93  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 6.73/6.93  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 6.73/6.93  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.73/6.93  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.73/6.93  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.73/6.93  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 6.73/6.93  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.73/6.93  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.73/6.93  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.73/6.93  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.73/6.93  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.73/6.93  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.73/6.93  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 6.73/6.93  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 6.73/6.93  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 6.73/6.93  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 6.73/6.93  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 6.73/6.93  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 6.73/6.93  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 6.73/6.93  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 6.73/6.93  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 6.73/6.93  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 6.73/6.93  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 6.73/6.93  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 6.73/6.93  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 6.73/6.93  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 6.73/6.93  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 6.73/6.93  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 6.73/6.93  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 7.04/7.22  Found ((eq_trans0000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 7.04/7.22  Found (((eq_trans000 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 7.04/7.22  Found ((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 7.04/7.22  Found (((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 7.04/7.22  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 7.04/7.22  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 7.04/7.22  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 7.04/7.22  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 7.04/7.22  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 7.04/7.22  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 7.04/7.22  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 7.04/7.22  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 7.04/7.22  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 7.04/7.22  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 7.04/7.22  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 7.04/7.22  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 7.04/7.22  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 7.04/7.22  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 7.04/7.22  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 7.04/7.22  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 7.04/7.22  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 7.04/7.22  Found ((eq_trans0000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 7.04/7.22  Found (((eq_trans000 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 7.04/7.22  Found ((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 7.04/7.22  Found (((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 7.04/7.22  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 7.04/7.22  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 7.04/7.22  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 7.34/7.51  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.34/7.51  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.34/7.51  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.34/7.51  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.34/7.51  Found (eq_sym1000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.34/7.51  Found (eq_sym1000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.34/7.51  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym100 x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.34/7.51  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym10 (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.34/7.51  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym1 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.34/7.51  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.34/7.51  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.34/7.51  Found eta_expansion000:=(eta_expansion00 (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))):(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) (fun (x:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 7.34/7.51  Found (eta_expansion00 (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 7.34/7.51  Found ((eta_expansion0 Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 7.34/7.51  Found (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 7.34/7.51  Found (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 7.34/7.51  Found (eq_sym0001 (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.34/7.51  Found (eq_sym0001 (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.34/7.51  Found ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.34/7.51  Found ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.50/7.66  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.50/7.66  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 7.50/7.66  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 7.50/7.66  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 7.50/7.66  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 7.50/7.66  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 7.50/7.66  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 7.50/7.66  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 7.50/7.66  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.50/7.66  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.50/7.66  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.50/7.66  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.50/7.66  Found ((eq_sym1000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.50/7.66  Found ((eq_sym1000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.50/7.66  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym100 x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.50/7.66  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym10 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.50/7.66  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym1 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.50/7.66  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.50/7.66  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.50/7.66  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 7.50/7.66  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 7.75/7.90  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 7.75/7.90  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 7.75/7.90  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 7.75/7.90  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 7.75/7.90  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 7.75/7.90  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 7.75/7.90  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 7.75/7.90  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 7.75/7.90  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 7.75/7.90  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 7.75/7.90  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 7.75/7.90  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 7.75/7.90  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 7.75/7.90  Found ((eq_trans0000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 7.75/7.90  Found (((eq_trans000 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 7.75/7.90  Found ((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 7.75/7.90  Found (((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 7.75/7.90  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 7.75/7.90  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 7.75/7.90  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.75/7.90  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.75/7.90  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.75/7.90  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.75/7.90  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.75/7.90  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.75/7.90  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.75/7.90  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.75/7.90  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.75/7.90  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.75/7.90  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.75/7.90  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 7.75/7.90  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.86/8.07  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.86/8.07  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.86/8.07  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.86/8.07  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.86/8.07  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.86/8.07  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.86/8.07  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.86/8.07  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.86/8.07  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.86/8.07  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.86/8.07  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 7.86/8.07  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 7.86/8.07  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 7.86/8.07  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 7.86/8.07  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 7.86/8.07  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 7.86/8.07  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.86/8.07  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.86/8.07  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.86/8.07  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 7.86/8.07  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.86/8.07  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.86/8.07  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.86/8.07  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.86/8.07  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 7.86/8.07  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.18/8.33  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.18/8.33  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 8.18/8.33  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 8.18/8.33  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 8.18/8.33  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 8.18/8.33  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 8.18/8.33  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 8.18/8.33  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 8.18/8.33  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 8.18/8.33  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 8.18/8.33  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 8.18/8.33  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.18/8.33  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.18/8.33  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.18/8.33  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.18/8.33  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.18/8.33  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.18/8.33  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.18/8.33  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 8.18/8.33  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 8.18/8.33  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 8.18/8.33  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 8.18/8.33  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 8.18/8.33  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 8.18/8.33  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.18/8.33  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.18/8.33  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.18/8.33  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.18/8.33  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 8.18/8.33  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((eq_trans0000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 8.33/8.48  Found (((eq_trans000 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 8.33/8.48  Found (((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 8.33/8.48  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 8.33/8.48  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 8.33/8.48  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 8.33/8.48  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 8.33/8.48  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 8.33/8.48  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 8.33/8.48  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.33/8.48  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 8.33/8.48  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((eq_trans0000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 8.33/8.48  Found (((eq_trans000 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 8.33/8.48  Found (((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 8.33/8.48  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 8.33/8.48  Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 8.33/8.48  Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 8.33/8.48  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 8.33/8.48  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 8.33/8.48  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:Prop), (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 8.36/8.52  Found (functional_extensionality_dep00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.36/8.52  Found (functional_extensionality_dep00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.36/8.52  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.36/8.52  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.36/8.52  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.36/8.52  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 8.36/8.52  Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 8.36/8.52  Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 8.36/8.52  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 8.36/8.52  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 8.36/8.52  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:Prop), (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 8.36/8.52  Found (functional_extensionality00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.36/8.52  Found (functional_extensionality00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.36/8.52  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.36/8.52  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.36/8.52  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.36/8.52  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 8.48/8.66  Found eta_expansion000:=(eta_expansion00 (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))):(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) (fun (x:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 8.48/8.66  Found (eta_expansion00 (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 8.48/8.66  Found ((eta_expansion0 Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 8.48/8.66  Found (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 8.48/8.66  Found (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 8.48/8.66  Found (eq_sym0001 (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.48/8.66  Found (eq_sym0001 (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.48/8.66  Found ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.48/8.66  Found ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.48/8.66  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.48/8.66  Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 8.48/8.66  Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 8.48/8.66  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 8.48/8.66  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 8.48/8.66  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:Prop), (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 8.48/8.66  Found (functional_extensionality00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.48/8.66  Found (functional_extensionality00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.48/8.66  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.57/8.77  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.57/8.77  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.57/8.77  Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 8.57/8.77  Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 8.57/8.77  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 8.57/8.77  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 8.57/8.77  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:Prop), (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 8.57/8.77  Found (functional_extensionality_dep00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.57/8.77  Found (functional_extensionality_dep00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.57/8.77  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.57/8.77  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.57/8.77  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.57/8.77  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 8.57/8.77  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 8.57/8.77  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 8.57/8.77  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 8.57/8.77  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 8.57/8.77  Found x0:(P0 (((eq Prop) x) Xy))
% 8.57/8.77  Instantiate: x:=Xy:Prop
% 8.57/8.77  Found (fun (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (P0 (((eq Prop) Xy) x))
% 8.57/8.77  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.57/8.77  Found (fun (P0:(Prop->Prop)) (x0:(P0 (((eq Prop) x) Xy)))=> x0) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 8.57/8.77  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 8.57/8.77  Found (eq_ref00 P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.57/8.77  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 8.57/8.77  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.25/9.45  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.25/9.45  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.25/9.45  Found (fun (P0:(Prop->Prop))=> (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.25/9.45  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 9.25/9.45  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.25/9.45  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.25/9.45  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.25/9.45  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.25/9.45  Found (eq_sym000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.25/9.45  Found ((eq_sym00 (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.25/9.45  Found (((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.25/9.45  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.25/9.45  Found ((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.25/9.45  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 9.25/9.45  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.25/9.45  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.25/9.45  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.25/9.45  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.25/9.45  Found (eq_sym1000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.25/9.45  Found (eq_sym1000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.25/9.45  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym100 x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.25/9.45  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym10 (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.25/9.45  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym1 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.25/9.45  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.25/9.45  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.25/9.45  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.25/9.45  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 9.35/9.57  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 9.35/9.57  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 9.35/9.57  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 9.35/9.57  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 9.35/9.57  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 9.35/9.57  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.35/9.57  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.35/9.57  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.35/9.57  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.35/9.57  Found ((eq_sym1000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.35/9.57  Found ((eq_sym1000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.35/9.57  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym100 x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.35/9.57  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym10 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.35/9.57  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym1 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.35/9.57  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.35/9.57  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.35/9.57  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.35/9.57  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 9.35/9.57  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 9.35/9.57  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 9.35/9.57  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 9.35/9.57  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 9.35/9.57  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 9.35/9.57  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 9.35/9.57  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 9.35/9.57  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 9.35/9.57  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 9.35/9.57  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 9.35/9.57  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 9.35/9.57  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 9.47/9.65  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 9.47/9.65  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 9.47/9.65  Found ((eq_trans0000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.47/9.65  Found (((eq_trans000 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.47/9.65  Found ((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.47/9.65  Found (((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.47/9.65  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.47/9.65  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.47/9.65  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 9.47/9.65  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.47/9.65  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.47/9.65  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.47/9.65  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.47/9.65  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.47/9.65  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.47/9.65  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.47/9.65  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.47/9.65  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.47/9.65  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.47/9.65  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.47/9.65  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.47/9.65  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 9.47/9.65  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.47/9.65  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.66/9.81  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.66/9.81  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.66/9.81  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.66/9.81  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.66/9.81  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.66/9.81  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.66/9.81  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.66/9.81  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.66/9.81  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.66/9.81  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.66/9.81  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 9.66/9.81  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 9.66/9.81  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 9.66/9.81  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 9.66/9.81  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 9.66/9.81  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 9.66/9.81  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.66/9.81  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.66/9.81  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.66/9.81  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.66/9.81  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.66/9.81  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.66/9.81  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.66/9.81  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.66/9.81  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.66/9.81  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.75/9.90  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.75/9.90  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 9.75/9.90  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 9.75/9.90  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 9.75/9.90  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 9.75/9.90  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 9.75/9.90  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 9.75/9.90  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 9.75/9.90  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.75/9.90  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.75/9.90  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.75/9.90  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 9.75/9.90  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.75/9.90  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.75/9.90  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.75/9.90  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.75/9.90  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.75/9.90  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.75/9.90  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 9.75/9.90  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.03/10.20  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 10.03/10.20  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 10.03/10.20  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 10.03/10.20  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 10.03/10.20  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 10.03/10.20  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 10.03/10.20  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 10.03/10.20  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_trans0000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.03/10.20  Found (((eq_trans000 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.03/10.20  Found ((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.03/10.20  Found (((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.03/10.20  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.03/10.20  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.03/10.20  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 10.03/10.20  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 10.03/10.20  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 10.03/10.20  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 10.03/10.20  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 10.03/10.20  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 10.03/10.20  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 10.03/10.20  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 10.03/10.20  Found ((eq_trans0000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.03/10.20  Found (((eq_trans000 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.03/10.20  Found ((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.13/10.29  Found (((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.13/10.29  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.13/10.29  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.13/10.29  Found eta_expansion000:=(eta_expansion00 (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))):(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) (fun (x:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 10.13/10.29  Found (eta_expansion00 (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 10.13/10.29  Found ((eta_expansion0 Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 10.13/10.29  Found (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 10.13/10.29  Found (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) as proof of (((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx))))) b)
% 10.13/10.29  Found (eq_sym0001 (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.13/10.29  Found (eq_sym0001 (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.13/10.29  Found ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.13/10.29  Found ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.13/10.29  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.13/10.29  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq (Prop->Prop)) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))) b))=> ((eq_sym000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.33/10.48  Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 10.33/10.48  Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 10.33/10.48  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 10.33/10.48  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 10.33/10.48  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:Prop), (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 10.33/10.48  Found (functional_extensionality_dep00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.48  Found (functional_extensionality_dep00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.48  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.48  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.48  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.48  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality_dep0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.33/10.48  Found eq_ref00:=(eq_ref0 (f x0)):(((eq Prop) (f x0)) (f x0))
% 10.33/10.48  Found (eq_ref0 (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 10.33/10.48  Found ((eq_ref Prop) (f x0)) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 10.33/10.48  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (((eq Prop) (f x0)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x0) Xy)) (((eq Prop) Xy) x0))))
% 10.33/10.48  Found (fun (x0:Prop)=> ((eq_ref Prop) (f x0))) as proof of (forall (x:Prop), (((eq Prop) (f x)) (forall (Xy:Prop), (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x)))))
% 10.33/10.48  Found (functional_extensionality00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.48  Found (functional_extensionality00001 (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.48  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.48  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.53  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.53  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x0) Xy0)) (((eq Prop) Xy0) x0))))))=> ((functional_extensionality0000 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop)=> ((eq_ref Prop) (f x0))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 10.33/10.53  Found x1:(P2 (f x0))
% 10.33/10.53  Instantiate: g:=f:(Prop->Prop)
% 10.33/10.53  Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (g x0))
% 10.33/10.53  Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (g x0)))
% 10.33/10.53  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (g x0))
% 10.33/10.53  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:Prop), (((eq Prop) (f x)) (g x)))
% 10.33/10.53  Found (functional_extensionality00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.53  Found (functional_extensionality00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.53  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (g x0))))=> ((functional_extensionality0001 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.53  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.53  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.53  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.53  Found x1:(P2 (f x0))
% 10.33/10.53  Instantiate: g:=f:(Prop->Prop)
% 10.33/10.53  Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (g x0))
% 10.33/10.53  Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (g x0)))
% 10.33/10.53  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (g x0))
% 10.33/10.53  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:Prop), (((eq Prop) (f x)) (g x)))
% 10.33/10.53  Found (functional_extensionality_dep00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.53  Found (functional_extensionality_dep00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.53  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (g x0))))=> ((functional_extensionality_dep0001 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.33/10.53  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.88/11.03  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.88/11.03  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.88/11.03  Found eq_ref00:=(eq_ref0 a):(((eq (Prop->Prop)) a) a)
% 10.88/11.03  Found (eq_ref0 a) as proof of (((eq (Prop->Prop)) a) b)
% 10.88/11.03  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b)
% 10.88/11.03  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b)
% 10.88/11.03  Found eq_ref00:=(eq_ref0 b):(((eq (Prop->Prop)) b) b)
% 10.88/11.03  Found (eq_ref0 b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 10.88/11.03  Found ((eq_ref (Prop->Prop)) b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 10.88/11.03  Found ((eq_ref (Prop->Prop)) b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 10.88/11.03  Found ((eq_trans00001 ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.88/11.03  Found ((eq_trans00001 ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.88/11.03  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.88/11.03  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.88/11.03  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.88/11.03  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 10.88/11.03  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 10.88/11.03  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 10.88/11.03  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 10.88/11.03  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 10.88/11.03  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.88/11.03  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.88/11.03  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.88/11.03  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 10.88/11.03  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.09/11.29  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.09/11.29  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.09/11.29  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 11.09/11.29  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 11.09/11.29  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 11.09/11.29  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 11.09/11.29  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 11.09/11.29  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 11.09/11.29  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 11.09/11.29  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 11.09/11.29  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 11.09/11.29  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 11.09/11.29  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 11.09/11.29  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.09/11.29  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.09/11.29  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.09/11.29  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.09/11.29  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.09/11.29  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.09/11.29  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.09/11.29  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 11.85/12.02  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 11.85/12.02  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 11.85/12.02  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 11.85/12.02  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 11.85/12.02  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 11.85/12.02  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 11.85/12.02  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 11.85/12.02  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found ((eq_trans1000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 11.85/12.02  Found (((eq_trans100 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 11.85/12.02  Found ((((eq_trans10 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 11.85/12.02  Found (((((eq_trans1 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 11.85/12.02  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 11.85/12.02  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 11.85/12.02  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 11.85/12.02  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 11.85/12.02  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 11.85/12.02  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 11.85/12.02  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 11.85/12.02  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 11.85/12.02  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 11.85/12.02  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 11.85/12.02  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 11.85/12.02  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 11.85/12.02  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 11.85/12.02  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 11.85/12.02  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.85/12.02  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 11.92/12.07  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.92/12.07  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.92/12.07  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.92/12.07  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.92/12.07  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.92/12.07  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.92/12.07  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.92/12.07  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 11.92/12.07  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 11.92/12.07  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 11.92/12.07  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 11.92/12.07  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 12.08/12.27  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 12.08/12.27  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 12.08/12.27  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 12.08/12.27  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 12.08/12.27  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 12.08/12.27  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 12.08/12.27  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 12.08/12.27  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 12.08/12.27  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 12.08/12.27  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 12.08/12.27  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 12.08/12.27  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 12.08/12.27  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.08/12.27  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.08/12.27  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.08/12.27  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.08/12.27  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.08/12.27  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.08/12.27  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.08/12.27  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.08/12.27  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 12.08/12.27  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 12.57/12.76  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 12.57/12.76  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 12.57/12.76  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 12.57/12.76  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 12.57/12.76  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.57/12.76  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.57/12.76  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.57/12.76  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.57/12.76  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.57/12.76  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.57/12.76  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.57/12.76  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 12.57/12.76  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 12.57/12.76  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 12.57/12.76  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 12.57/12.76  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 12.57/12.76  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 12.57/12.76  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 12.57/12.76  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 12.57/12.76  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 12.57/12.76  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 12.57/12.76  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 12.57/12.76  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 12.57/12.76  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 12.57/12.76  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 12.57/12.76  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 12.57/12.76  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 12.57/12.76  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 12.57/12.76  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 12.57/12.76  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 12.57/12.76  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 12.57/12.76  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.57/12.76  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.57/12.76  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.57/12.76  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.76/12.94  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.76/12.94  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.76/12.94  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.76/12.94  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 12.76/12.94  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 12.76/12.94  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 12.76/12.94  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 12.76/12.94  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 12.76/12.94  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 12.76/12.94  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.76/12.94  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.76/12.94  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.76/12.94  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.76/12.94  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 12.76/12.94  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.76/12.94  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.76/12.94  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.76/12.94  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.76/12.94  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.76/12.94  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.76/12.94  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.76/12.94  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.76/12.94  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.76/12.94  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.96/13.10  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.96/13.10  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.96/13.10  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 12.96/13.10  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 12.96/13.10  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 12.96/13.10  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 12.96/13.10  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 12.96/13.10  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 12.96/13.10  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 12.96/13.10  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.96/13.10  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.96/13.10  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.96/13.10  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.96/13.10  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 12.96/13.10  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.96/13.10  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.96/13.10  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.96/13.10  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 12.96/13.10  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.96/13.10  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.96/13.10  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.96/13.10  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 12.96/13.10  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.06/13.25  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.06/13.25  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.06/13.25  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.06/13.25  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 13.06/13.25  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 13.06/13.25  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 13.06/13.25  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 13.06/13.25  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 13.06/13.25  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 13.06/13.25  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 13.06/13.25  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.06/13.25  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.06/13.25  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.06/13.25  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.06/13.25  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 13.06/13.25  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.06/13.25  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.06/13.25  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.06/13.25  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.06/13.25  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.06/13.25  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.06/13.25  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.06/13.25  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.27/13.41  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.27/13.41  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.27/13.41  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.27/13.41  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.27/13.41  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 13.27/13.41  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 13.27/13.41  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 13.27/13.41  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 13.27/13.41  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 13.27/13.41  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 13.27/13.41  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 13.27/13.41  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.27/13.41  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.27/13.41  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.27/13.41  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.27/13.41  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 13.27/13.41  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.27/13.41  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.27/13.41  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.38/13.53  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 13.38/13.53  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.38/13.53  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.38/13.53  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.38/13.53  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.38/13.53  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.38/13.53  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.38/13.53  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.38/13.53  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.38/13.53  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 13.38/13.53  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 13.38/13.53  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 13.38/13.53  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 13.38/13.53  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 13.38/13.53  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 13.38/13.53  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 13.38/13.53  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.38/13.53  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.38/13.53  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.38/13.53  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.38/13.53  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 13.38/13.53  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.38/13.53  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.86/14.05  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.86/14.05  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.86/14.05  Found ((eq_trans1000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 13.86/14.05  Found (((eq_trans100 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 13.86/14.05  Found ((((eq_trans10 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 13.86/14.05  Found (((((eq_trans1 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 13.86/14.05  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) Xy) x))
% 13.86/14.05  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 13.86/14.05  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 13.86/14.05  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 13.86/14.05  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 13.86/14.05  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 13.86/14.05  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 13.86/14.05  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 13.86/14.05  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 13.86/14.05  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 13.86/14.05  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 13.86/14.05  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 13.86/14.05  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.86/14.05  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.86/14.05  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.86/14.05  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.86/14.05  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 13.86/14.05  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.86/14.05  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.86/14.05  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.86/14.05  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 13.86/14.05  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.86/14.05  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.86/14.05  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.86/14.05  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.86/14.05  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 13.86/14.05  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.07/14.23  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.07/14.23  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.07/14.23  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 14.07/14.23  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 14.07/14.23  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 14.07/14.23  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 14.07/14.23  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 14.07/14.23  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 14.07/14.23  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 14.07/14.23  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 14.07/14.23  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 14.07/14.23  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 14.07/14.23  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 14.07/14.23  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 14.07/14.23  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 14.07/14.23  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 14.07/14.23  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 14.07/14.23  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.07/14.23  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.07/14.23  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.07/14.23  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.07/14.23  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.07/14.23  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.53/14.69  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.53/14.69  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.53/14.69  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 14.53/14.69  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 14.53/14.69  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 14.53/14.69  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 14.53/14.69  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 14.53/14.69  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 14.53/14.69  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 14.53/14.69  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 14.53/14.69  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 14.53/14.69  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 14.53/14.69  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 14.53/14.69  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.53/14.69  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.53/14.69  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.53/14.69  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.53/14.69  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 14.53/14.69  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.53/14.69  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.53/14.69  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.53/14.69  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.53/14.69  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 14.53/14.69  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.53/14.69  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.53/14.69  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.53/14.69  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.53/14.69  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.53/14.69  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.53/14.69  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.53/14.69  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.69/14.90  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.69/14.90  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.69/14.90  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.69/14.90  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.69/14.90  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 14.69/14.90  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.69/14.90  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.69/14.90  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.69/14.90  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.69/14.90  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 14.69/14.90  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.69/14.90  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.69/14.90  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.69/14.90  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.69/14.90  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 14.69/14.90  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.69/14.90  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.69/14.90  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.69/14.90  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.69/14.90  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.69/14.90  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.69/14.90  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.69/14.90  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.69/14.90  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.82/15.04  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.82/15.04  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.82/15.04  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.82/15.04  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 14.82/15.04  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.82/15.04  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.82/15.04  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.82/15.04  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.82/15.04  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 14.82/15.04  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.82/15.04  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.82/15.04  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.82/15.04  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.82/15.04  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 14.82/15.04  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.82/15.04  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.82/15.04  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.82/15.04  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.82/15.04  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.82/15.04  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.82/15.04  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.82/15.04  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.82/15.04  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.19  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.19  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.19  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.19  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 14.98/15.19  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.98/15.19  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.98/15.19  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.98/15.19  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 14.98/15.19  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 14.98/15.19  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.98/15.19  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.98/15.19  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.98/15.19  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.98/15.19  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 14.98/15.19  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.98/15.19  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.98/15.19  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.98/15.19  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 14.98/15.19  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.19  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.19  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.19  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.19  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.20  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.20  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.20  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.20  Found x1:(P2 (f x0))
% 14.98/15.20  Instantiate: g:=f:(Prop->Prop)
% 14.98/15.20  Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (g x0))
% 14.98/15.20  Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (g x0)))
% 14.98/15.20  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (g x0))
% 14.98/15.20  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:Prop), (((eq Prop) (f x)) (g x)))
% 14.98/15.20  Found (functional_extensionality00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.20  Found (functional_extensionality00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.20  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (g x0))))=> ((functional_extensionality0001 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.20  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.20  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.20  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 14.98/15.20  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 15.14/15.34  Found x1:(P2 (f x0))
% 15.14/15.34  Instantiate: g:=f:(Prop->Prop)
% 15.14/15.34  Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (g x0))
% 15.14/15.34  Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (g x0)))
% 15.14/15.34  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (g x0))
% 15.14/15.34  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:Prop), (((eq Prop) (f x)) (g x)))
% 15.14/15.34  Found (functional_extensionality_dep00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.14/15.34  Found (functional_extensionality_dep00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.14/15.34  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (g x0))))=> ((functional_extensionality_dep0001 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.14/15.34  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.14/15.34  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.14/15.34  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.14/15.34  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 15.14/15.34  Found x1:(P2 (f x0))
% 15.14/15.34  Instantiate: g:=f:(Prop->Prop)
% 15.14/15.34  Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (g x0))
% 15.14/15.34  Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (g x0)))
% 15.14/15.34  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (g x0))
% 15.14/15.34  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:Prop), (((eq Prop) (f x)) (g x)))
% 15.14/15.34  Found (functional_extensionality00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.14/15.34  Found (functional_extensionality00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.14/15.34  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (g x0))))=> ((functional_extensionality0001 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.14/15.34  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.14/15.34  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.14/15.34  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.28/15.50  Found x1:(P2 (f x0))
% 15.28/15.50  Instantiate: g:=f:(Prop->Prop)
% 15.28/15.50  Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (g x0))
% 15.28/15.50  Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (g x0)))
% 15.28/15.50  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (g x0))
% 15.28/15.50  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:Prop), (((eq Prop) (f x)) (g x)))
% 15.28/15.50  Found (functional_extensionality_dep00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.28/15.50  Found (functional_extensionality_dep00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.28/15.50  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (g x0))))=> ((functional_extensionality_dep0001 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.28/15.50  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.28/15.50  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.28/15.50  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.28/15.50  Found eq_ref00:=(eq_ref0 b):(((eq (Prop->Prop)) b) b)
% 15.28/15.50  Found (eq_ref0 b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 15.28/15.50  Found ((eq_ref (Prop->Prop)) b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 15.28/15.50  Found ((eq_ref (Prop->Prop)) b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 15.28/15.50  Found eq_ref00:=(eq_ref0 a):(((eq (Prop->Prop)) a) a)
% 15.28/15.50  Found (eq_ref0 a) as proof of (((eq (Prop->Prop)) a) b)
% 15.28/15.50  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b)
% 15.28/15.50  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b)
% 15.28/15.50  Found ((eq_trans00001 ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.28/15.50  Found ((eq_trans00001 ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.28/15.50  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.28/15.50  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.28/15.50  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.68/15.85  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 15.68/15.85  Found eq_ref00:=(eq_ref0 a):(((eq (Prop->Prop)) a) a)
% 15.68/15.85  Found (eq_ref0 a) as proof of (((eq (Prop->Prop)) a) b)
% 15.68/15.85  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b)
% 15.68/15.85  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b)
% 15.68/15.85  Found eq_ref00:=(eq_ref0 b):(((eq (Prop->Prop)) b) b)
% 15.68/15.85  Found (eq_ref0 b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 15.68/15.85  Found ((eq_ref (Prop->Prop)) b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 15.68/15.85  Found ((eq_ref (Prop->Prop)) b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 15.68/15.85  Found ((eq_trans00001 ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.68/15.85  Found ((eq_trans00001 ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.68/15.85  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.68/15.85  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.68/15.85  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.68/15.85  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 15.68/15.85  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 15.68/15.85  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 15.68/15.85  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 15.68/15.85  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 15.68/15.85  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.68/15.85  Found (eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.68/15.85  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.68/15.85  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.68/15.85  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.68/15.85  Found ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.96/16.11  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.96/16.11  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) P0)) ((eq_ref Prop) (((eq Prop) Xy) x)))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 15.96/16.11  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 15.96/16.11  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 15.96/16.11  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 15.96/16.11  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 15.96/16.11  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 15.96/16.11  Found eq_ref00:=(eq_ref0 (((eq Prop) Xy) x)):(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) Xy) x))
% 15.96/16.11  Found (eq_ref0 (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 15.96/16.11  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 15.96/16.11  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 15.96/16.11  Found ((eq_ref Prop) (((eq Prop) Xy) x)) as proof of (((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy))
% 15.96/16.11  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.96/16.11  Found ((eq_sym0000 ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.96/16.11  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((eq_sym000 x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.96/16.11  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((eq_sym00 (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.96/16.11  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> ((((eq_sym0 (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.96/16.11  Found (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.96/16.11  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 15.96/16.11  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)))=> (((((eq_sym Prop) (((eq Prop) Xy) x)) (((eq Prop) x) Xy)) x0) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) Xy) x))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 15.96/16.11  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 15.96/16.11  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 16.56/16.74  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 16.56/16.74  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 16.56/16.74  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 16.56/16.74  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 16.56/16.74  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 16.56/16.74  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_trans1000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 16.56/16.74  Found (((eq_trans100 (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 16.56/16.74  Found ((((eq_trans10 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 16.56/16.74  Found (((((eq_trans1 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 16.56/16.74  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 16.56/16.74  Found ((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 16.56/16.74  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 16.56/16.74  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 16.56/16.74  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 16.56/16.74  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 16.56/16.74  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 16.56/16.74  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 16.56/16.74  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 16.56/16.74  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 16.56/16.74  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 16.56/16.74  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 16.56/16.74  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 16.56/16.74  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 16.56/16.74  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.56/16.74  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.56/16.74  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.92  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.92  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.92  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.92  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.92  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.92  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.92  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 16.75/16.92  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 16.75/16.92  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 16.75/16.92  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 16.75/16.92  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 16.75/16.92  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 16.75/16.92  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 16.75/16.92  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.75/16.92  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.75/16.92  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.75/16.92  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.75/16.92  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 16.75/16.92  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.75/16.92  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.75/16.92  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.75/16.92  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 16.75/16.98  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.98  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.98  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.98  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.98  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.98  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.98  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.98  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 16.75/16.98  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 16.75/16.98  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 16.75/16.98  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 16.75/16.98  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 16.75/16.98  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 17.17/17.32  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 17.17/17.32  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 17.17/17.32  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 17.17/17.32  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 17.17/17.32  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 17.17/17.32  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 17.17/17.32  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 17.17/17.32  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.17/17.32  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.17/17.32  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.17/17.32  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.17/17.32  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 17.17/17.32  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.17/17.32  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.17/17.32  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.17/17.32  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.17/17.32  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 17.17/17.32  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.17/17.32  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.17/17.32  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.17/17.32  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.17/17.32  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.17/17.32  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.17/17.32  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.17/17.32  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.17/17.32  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.17/17.32  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.17/17.32  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.17/17.32  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.35/17.51  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 17.35/17.51  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 17.35/17.51  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.35/17.51  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.35/17.51  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.35/17.51  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.35/17.51  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 17.35/17.51  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.35/17.51  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.35/17.51  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.35/17.51  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.35/17.51  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 17.35/17.51  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.35/17.51  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.35/17.51  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.35/17.51  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.35/17.51  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.35/17.51  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.35/17.51  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.35/17.51  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.35/17.51  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.35/17.51  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.35/17.51  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.48/17.70  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.48/17.70  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 17.48/17.70  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 17.48/17.70  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.48/17.70  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.48/17.70  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.48/17.70  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.48/17.70  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 17.48/17.70  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.48/17.70  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.48/17.70  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.48/17.70  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.48/17.70  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 17.48/17.70  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.48/17.70  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.48/17.70  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.48/17.70  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.48/17.70  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.48/17.70  Found ((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.48/17.70  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.48/17.70  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.48/17.70  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.48/17.70  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.72/17.86  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.72/17.86  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.72/17.86  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 17.72/17.86  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 17.72/17.86  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.72/17.86  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.72/17.86  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.72/17.86  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b)
% 17.72/17.86  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 17.72/17.86  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.72/17.86  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.72/17.86  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.72/17.86  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.72/17.86  Found eq_ref00:=(eq_ref0 b):(((eq Prop) b) b)
% 17.72/17.86  Found (eq_ref0 b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.72/17.86  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.72/17.86  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.72/17.86  Found ((eq_ref Prop) b) as proof of (((eq Prop) b) (((eq Prop) Xy) x))
% 17.72/17.86  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.72/17.86  Found (((eq_trans00000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.72/17.86  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> (((eq_trans0000 x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.72/17.86  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b)) (x00:(((eq Prop) b) (((eq Prop) Xy) x)))=> ((((eq_trans000 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.72/17.86  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans00 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.79/17.99  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans0 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.79/17.99  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.79/17.99  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.79/17.99  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 17.79/17.99  Found x1:(P2 (f x0))
% 17.79/17.99  Instantiate: g:=f:(Prop->Prop)
% 17.79/17.99  Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (g x0))
% 17.79/17.99  Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (g x0)))
% 17.79/17.99  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (g x0))
% 17.79/17.99  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:Prop), (((eq Prop) (f x)) (g x)))
% 17.79/17.99  Found (functional_extensionality00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.79/17.99  Found (functional_extensionality00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.79/17.99  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (g x0))))=> ((functional_extensionality0001 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.79/17.99  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.79/17.99  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 17.79/17.99  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.09/18.24  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 18.09/18.24  Found x1:(P2 (f x0))
% 18.09/18.24  Instantiate: g:=f:(Prop->Prop)
% 18.09/18.24  Found (fun (x1:(P2 (f x0)))=> x1) as proof of (P2 (g x0))
% 18.09/18.24  Found (fun (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of ((P2 (f x0))->(P2 (g x0)))
% 18.09/18.24  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (((eq Prop) (f x0)) (g x0))
% 18.09/18.24  Found (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1) as proof of (forall (x:Prop), (((eq Prop) (f x)) (g x)))
% 18.09/18.24  Found (functional_extensionality_dep00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.09/18.24  Found (functional_extensionality_dep00010 (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.09/18.24  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (g x0))))=> ((functional_extensionality_dep0001 x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.09/18.24  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.09/18.24  Found ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.09/18.24  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.09/18.24  Found (fun (P0:(Prop->Prop))=> ((fun (x0:(forall (x0:Prop), (((eq Prop) (f x0)) (f x0))))=> (((functional_extensionality_dep000 f) x0) (fun (x1:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (fun (x0:Prop) (P2:(Prop->Prop)) (x1:(P2 (f x0)))=> x1))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 18.09/18.24  Found eq_ref00:=(eq_ref0 b):(((eq (Prop->Prop)) b) b)
% 18.09/18.24  Found (eq_ref0 b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 18.09/18.24  Found ((eq_ref (Prop->Prop)) b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 18.09/18.24  Found ((eq_ref (Prop->Prop)) b) as proof of (((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy:Prop), (((eq Prop) (((eq Prop) Xx) Xy)) (((eq Prop) Xy) Xx)))))
% 18.09/18.24  Found eq_ref00:=(eq_ref0 a):(((eq (Prop->Prop)) a) a)
% 18.09/18.24  Found (eq_ref0 a) as proof of (((eq (Prop->Prop)) a) b)
% 18.09/18.24  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b)
% 18.09/18.24  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b)
% 18.09/18.24  Found ((eq_trans00001 ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.09/18.24  Found ((eq_trans00001 ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.09/18.24  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.46/18.64  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.46/18.64  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.46/18.64  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (Xx:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) Xx) Xy0)) (((eq Prop) Xy0) Xx))))))=> (((eq_trans0000 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) ((eq_ref (Prop->Prop)) b))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 18.46/18.64  Found eta_expansion000:=(eta_expansion00 a):(((eq (Prop->Prop)) a) (fun (x:Prop)=> (a x)))
% 18.46/18.64  Found (eta_expansion00 a) as proof of (((eq (Prop->Prop)) a) b)
% 18.46/18.64  Found ((eta_expansion0 Prop) a) as proof of (((eq (Prop->Prop)) a) b)
% 18.46/18.64  Found (((eta_expansion Prop) Prop) a) as proof of (((eq (Prop->Prop)) a) b)
% 18.46/18.64  Found (((eta_expansion Prop) Prop) a) as proof of (((eq (Prop->Prop)) a) b)
% 18.46/18.64  Found eta_expansion000:=(eta_expansion00 b):(((eq (Prop->Prop)) b) (fun (x:Prop)=> (b x)))
% 18.46/18.64  Found (eta_expansion00 b) as proof of (((eq (Prop->Prop)) b) c)
% 18.46/18.64  Found ((eta_expansion0 Prop) b) as proof of (((eq (Prop->Prop)) b) c)
% 18.46/18.64  Found (((eta_expansion Prop) Prop) b) as proof of (((eq (Prop->Prop)) b) c)
% 18.46/18.64  Found (((eta_expansion Prop) Prop) b) as proof of (((eq (Prop->Prop)) b) c)
% 18.46/18.64  Found (((eta_expansion Prop) Prop) b) as proof of (((eq (Prop->Prop)) b) c)
% 18.46/18.64  Found ((eq_trans00010 (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.46/18.64  Found ((eq_trans00010 (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.46/18.64  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) c))=> (((eq_trans0001 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.46/18.64  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.46/18.64  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.46/18.64  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.46/18.64  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 18.68/18.86  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 18.68/18.86  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 18.68/18.86  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 18.68/18.86  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 18.68/18.86  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 18.68/18.86  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 18.68/18.86  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 18.68/18.86  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 18.68/18.86  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 18.68/18.86  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 18.68/18.86  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.68/18.86  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.68/18.86  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.68/18.86  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.68/18.86  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 18.68/18.86  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.68/18.86  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.68/18.86  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.68/18.86  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.68/18.86  Found ((eq_trans10000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.68/18.86  Found ((eq_trans10000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.68/18.86  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> (((eq_trans1000 x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.68/18.86  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> ((((eq_trans100 (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.68/18.86  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans10 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.68/18.86  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans1 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.68/18.86  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.68/18.86  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.84/19.01  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 18.84/19.01  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 18.84/19.01  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 18.84/19.01  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 18.84/19.01  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 18.84/19.01  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 18.84/19.01  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 18.84/19.01  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.84/19.01  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.84/19.01  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.84/19.01  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.84/19.01  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 18.84/19.01  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.84/19.01  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.84/19.01  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.84/19.01  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 18.84/19.01  Found (((eq_trans10000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.84/19.01  Found (((eq_trans10000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.84/19.01  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> (((eq_trans1000 x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.84/19.01  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> ((((eq_trans100 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.84/19.01  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans10 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.84/19.01  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans1 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 18.84/19.01  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.25/19.40  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.25/19.40  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 19.25/19.40  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 19.25/19.40  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 19.25/19.40  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 19.25/19.40  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 19.25/19.40  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 19.25/19.40  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 19.25/19.40  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 19.25/19.40  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 19.25/19.40  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 19.25/19.40  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 19.25/19.40  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 19.25/19.40  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.25/19.40  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.25/19.40  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.25/19.40  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.25/19.40  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 19.25/19.40  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.25/19.40  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.25/19.40  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.25/19.40  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.25/19.40  Found ((eq_trans10000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.25/19.40  Found ((eq_trans10000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.25/19.40  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> (((eq_trans1000 x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.25/19.40  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> ((((eq_trans100 (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.25/19.40  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans10 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.45/19.63  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans1 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.45/19.63  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.45/19.63  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.45/19.63  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 19.45/19.63  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 19.45/19.63  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 19.45/19.63  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 19.45/19.63  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 19.45/19.63  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 19.45/19.63  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.45/19.63  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.45/19.63  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.45/19.63  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.45/19.63  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 19.45/19.63  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.45/19.63  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.45/19.63  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.45/19.63  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 19.45/19.63  Found (((eq_trans10000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.45/19.63  Found (((eq_trans10000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.45/19.63  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> (((eq_trans1000 x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.45/19.63  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> ((((eq_trans100 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.45/19.63  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans10 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.68/19.87  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans1 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.68/19.87  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.68/19.87  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.68/19.87  Found eta_expansion000:=(eta_expansion00 a):(((eq (Prop->Prop)) a) (fun (x:Prop)=> (a x)))
% 19.68/19.87  Found (eta_expansion00 a) as proof of (((eq (Prop->Prop)) a) b)
% 19.68/19.87  Found ((eta_expansion0 Prop) a) as proof of (((eq (Prop->Prop)) a) b)
% 19.68/19.87  Found (((eta_expansion Prop) Prop) a) as proof of (((eq (Prop->Prop)) a) b)
% 19.68/19.87  Found (((eta_expansion Prop) Prop) a) as proof of (((eq (Prop->Prop)) a) b)
% 19.68/19.87  Found eta_expansion000:=(eta_expansion00 b):(((eq (Prop->Prop)) b) (fun (x:Prop)=> (b x)))
% 19.68/19.87  Found (eta_expansion00 b) as proof of (((eq (Prop->Prop)) b) c)
% 19.68/19.87  Found ((eta_expansion0 Prop) b) as proof of (((eq (Prop->Prop)) b) c)
% 19.68/19.87  Found (((eta_expansion Prop) Prop) b) as proof of (((eq (Prop->Prop)) b) c)
% 19.68/19.87  Found (((eta_expansion Prop) Prop) b) as proof of (((eq (Prop->Prop)) b) c)
% 19.68/19.87  Found (((eta_expansion Prop) Prop) b) as proof of (((eq (Prop->Prop)) b) c)
% 19.68/19.87  Found ((eq_trans00010 (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.68/19.87  Found ((eq_trans00010 (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.68/19.87  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) c))=> (((eq_trans0001 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.68/19.87  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.68/19.87  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.68/19.87  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.96/20.19  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 19.96/20.19  Found eta_expansion000:=(eta_expansion00 a):(((eq (Prop->Prop)) a) (fun (x:Prop)=> (a x)))
% 19.96/20.19  Found (eta_expansion00 a) as proof of (((eq (Prop->Prop)) a) b)
% 19.96/20.19  Found ((eta_expansion0 Prop) a) as proof of (((eq (Prop->Prop)) a) b)
% 19.96/20.19  Found (((eta_expansion Prop) Prop) a) as proof of (((eq (Prop->Prop)) a) b)
% 19.96/20.19  Found (((eta_expansion Prop) Prop) a) as proof of (((eq (Prop->Prop)) a) b)
% 19.96/20.19  Found eta_expansion_dep000:=(eta_expansion_dep00 b):(((eq (Prop->Prop)) b) (fun (x:Prop)=> (b x)))
% 19.96/20.19  Found (eta_expansion_dep00 b) as proof of (((eq (Prop->Prop)) b) c)
% 19.96/20.19  Found ((eta_expansion_dep0 (fun (x1:Prop)=> Prop)) b) as proof of (((eq (Prop->Prop)) b) c)
% 19.96/20.19  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b) as proof of (((eq (Prop->Prop)) b) c)
% 19.96/20.19  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b) as proof of (((eq (Prop->Prop)) b) c)
% 19.96/20.19  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b) as proof of (((eq (Prop->Prop)) b) c)
% 19.96/20.19  Found ((eq_trans00010 (((eta_expansion Prop) Prop) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.96/20.19  Found ((eq_trans00010 (((eta_expansion Prop) Prop) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.96/20.19  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) c))=> (((eq_trans0001 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.96/20.19  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.96/20.19  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.96/20.19  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 19.96/20.19  Found eq_ref000:=(eq_ref00 P0):((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) x) Xy)))
% 20.27/20.47  Found (eq_ref00 P0) as proof of (P1 (((eq Prop) x) Xy))
% 20.27/20.47  Found ((eq_ref0 (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 20.27/20.47  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 20.27/20.47  Found (((eq_ref Prop) (((eq Prop) x) Xy)) P0) as proof of (P1 (((eq Prop) x) Xy))
% 20.27/20.47  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 20.27/20.47  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 20.27/20.47  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 20.27/20.47  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 20.27/20.47  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 20.27/20.47  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 20.27/20.47  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.27/20.47  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.27/20.47  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.27/20.47  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.27/20.47  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 20.27/20.47  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.27/20.47  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.27/20.47  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.27/20.47  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.27/20.47  Found (((eq_trans10000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.27/20.47  Found (((eq_trans10000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.27/20.47  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> (((eq_trans1000 x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.27/20.47  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> ((((eq_trans100 (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.27/20.47  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans10 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.27/20.47  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans1 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.27/20.47  Found ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.45/20.61  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.45/20.61  Found (fun (P0:(Prop->Prop))=> ((((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) (fun (x1:Prop)=> ((P0 (((eq Prop) x) Xy))->(P0 x1))))) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (((eq_ref Prop) (((eq Prop) x) Xy)) P0))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 20.45/20.61  Found eq_ref00:=(eq_ref0 (((eq Prop) x) Xy)):(((eq Prop) (((eq Prop) x) Xy)) (((eq Prop) x) Xy))
% 20.45/20.61  Found (eq_ref0 (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 20.45/20.61  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 20.45/20.61  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 20.45/20.61  Found ((eq_ref Prop) (((eq Prop) x) Xy)) as proof of (((eq Prop) (((eq Prop) x) Xy)) b0)
% 20.45/20.61  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 20.45/20.61  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.45/20.61  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.45/20.61  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.45/20.61  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.45/20.61  Found eq_ref00:=(eq_ref0 b0):(((eq Prop) b0) b0)
% 20.45/20.61  Found (eq_ref0 b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.45/20.61  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.45/20.61  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.45/20.61  Found ((eq_ref Prop) b0) as proof of (((eq Prop) b0) (((eq Prop) Xy) x))
% 20.45/20.61  Found ((eq_trans10000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.45/20.61  Found ((eq_trans10000 ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.45/20.61  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> (((eq_trans1000 x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.45/20.61  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) b0)) (x00:(((eq Prop) b0) (((eq Prop) Xy) x)))=> ((((eq_trans100 (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.45/20.61  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((eq_trans10 (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.45/20.61  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> ((((((eq_trans1 (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.45/20.61  Found (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.58/20.79  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.58/20.79  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy))))) (x00:(((eq Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)))=> (((((((eq_trans Prop) (((eq Prop) x) Xy)) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))) (((eq Prop) Xy) x)) x0) x00) P0)) ((eq_ref Prop) (((eq Prop) x) Xy))) ((eq_ref Prop) (forall (P:(Prop->Prop)), ((P x)->(P Xy)))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 20.58/20.79  Found eta_expansion000:=(eta_expansion00 a):(((eq (Prop->Prop)) a) (fun (x:Prop)=> (a x)))
% 20.58/20.79  Found (eta_expansion00 a) as proof of (((eq (Prop->Prop)) a) b)
% 20.58/20.79  Found ((eta_expansion0 Prop) a) as proof of (((eq (Prop->Prop)) a) b)
% 20.58/20.79  Found (((eta_expansion Prop) Prop) a) as proof of (((eq (Prop->Prop)) a) b)
% 20.58/20.79  Found (((eta_expansion Prop) Prop) a) as proof of (((eq (Prop->Prop)) a) b)
% 20.58/20.79  Found eta_expansion000:=(eta_expansion00 b):(((eq (Prop->Prop)) b) (fun (x:Prop)=> (b x)))
% 20.58/20.79  Found (eta_expansion00 b) as proof of (((eq (Prop->Prop)) b) c)
% 20.58/20.79  Found ((eta_expansion0 Prop) b) as proof of (((eq (Prop->Prop)) b) c)
% 20.58/20.79  Found (((eta_expansion Prop) Prop) b) as proof of (((eq (Prop->Prop)) b) c)
% 20.58/20.79  Found (((eta_expansion Prop) Prop) b) as proof of (((eq (Prop->Prop)) b) c)
% 20.58/20.79  Found (((eta_expansion Prop) Prop) b) as proof of (((eq (Prop->Prop)) b) c)
% 20.58/20.79  Found ((eq_trans00010 (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.58/20.79  Found ((eq_trans00010 (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.58/20.79  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) c))=> (((eq_trans0001 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.58/20.79  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.58/20.79  Found (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.58/20.79  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.97/21.18  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) b)) (x1:(((eq (Prop->Prop)) b) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))=> ((((eq_trans000 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion Prop) Prop) b))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 20.97/21.18  Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (Prop->Prop)) a) (fun (x:Prop)=> (a x)))
% 20.97/21.18  Found (eta_expansion_dep00 a) as proof of (((eq (Prop->Prop)) a) b0)
% 20.97/21.18  Found ((eta_expansion_dep0 (fun (x1:Prop)=> Prop)) a) as proof of (((eq (Prop->Prop)) a) b0)
% 20.97/21.18  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a) as proof of (((eq (Prop->Prop)) a) b0)
% 20.97/21.18  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a) as proof of (((eq (Prop->Prop)) a) b0)
% 20.97/21.18  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a) as proof of (((eq (Prop->Prop)) a) b0)
% 20.97/21.18  Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (Prop->Prop)) b0) (fun (x:Prop)=> (b0 x)))
% 20.97/21.18  Found (eta_expansion_dep00 b0) as proof of (((eq (Prop->Prop)) b0) c)
% 20.97/21.18  Found ((eta_expansion_dep0 (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 20.97/21.18  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 20.97/21.18  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 20.97/21.18  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 20.97/21.18  Found ((eq_trans00100 (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.97/21.18  Found ((eq_trans00100 (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.97/21.18  Found (((fun (x0:(((eq (Prop->Prop)) a) b0)) (x1:(((eq (Prop->Prop)) b0) c))=> (((eq_trans0010 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.97/21.18  Found (((fun (x0:(((eq (Prop->Prop)) a) b0)) (x1:(((eq (Prop->Prop)) b0) (fun (x2:Prop)=> (b0 x2))))=> ((((eq_trans001 (fun (x2:Prop)=> (b0 x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.97/21.18  Found (((fun (x0:(((eq (Prop->Prop)) a) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) (x1:(((eq (Prop->Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))))=> (((((eq_trans00 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 20.97/21.18  Found (((fun (x0:(((eq (Prop->Prop)) a) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) (x1:(((eq (Prop->Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))))=> (((((eq_trans00 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.33/21.53  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) (x1:(((eq (Prop->Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))))=> (((((eq_trans00 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.33/21.53  Found eq_ref00:=(eq_ref0 a):(((eq (Prop->Prop)) a) a)
% 21.33/21.53  Found (eq_ref0 a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.33/21.53  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.33/21.53  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.33/21.53  Found ((eq_ref (Prop->Prop)) a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.33/21.53  Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (Prop->Prop)) b0) (fun (x:Prop)=> (b0 x)))
% 21.33/21.53  Found (eta_expansion_dep00 b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.33/21.53  Found ((eta_expansion_dep0 (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.33/21.53  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.33/21.53  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.33/21.53  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.33/21.53  Found ((eq_trans00100 ((eq_ref (Prop->Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.33/21.53  Found ((eq_trans00100 ((eq_ref (Prop->Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.33/21.53  Found (((fun (x0:(((eq (Prop->Prop)) a) b0)) (x1:(((eq (Prop->Prop)) b0) c))=> (((eq_trans0010 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.33/21.53  Found (((fun (x0:(((eq (Prop->Prop)) a) b0)) (x1:(((eq (Prop->Prop)) b0) (fun (x2:Prop)=> (b0 x2))))=> ((((eq_trans001 (fun (x2:Prop)=> (b0 x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.33/21.53  Found (((fun (x0:(((eq (Prop->Prop)) a) (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))))) (x1:(((eq (Prop->Prop)) (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3))))) (fun (x2:Prop)=> ((fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))) x2))))=> (((((eq_trans00 (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3))))) (fun (x2:Prop)=> ((fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))) x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.56/21.76  Found (((fun (x0:(((eq (Prop->Prop)) a) (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))))) (x1:(((eq (Prop->Prop)) (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3))))) (fun (x2:Prop)=> ((fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))) x2))))=> (((((eq_trans00 (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3))))) (fun (x2:Prop)=> ((fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))) x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.56/21.76  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))))) (x1:(((eq (Prop->Prop)) (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3))))) (fun (x2:Prop)=> ((fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))) x2))))=> (((((eq_trans00 (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3))))) (fun (x2:Prop)=> ((fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))) x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3))))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.56/21.76  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))))) (x1:(((eq (Prop->Prop)) (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3))))) (fun (x2:Prop)=> ((fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))) x2))))=> (((((eq_trans00 (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3))))) (fun (x2:Prop)=> ((fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3)))) x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) ((eq_ref (Prop->Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) (fun (x3:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x3) Xy0)) (((eq Prop) Xy0) x3))))))) as proof of (forall (P:(Prop->Prop)), ((P (((eq Prop) x) Xy))->(P (((eq Prop) Xy) x))))
% 21.56/21.76  Found eta_expansion_dep000:=(eta_expansion_dep00 a):(((eq (Prop->Prop)) a) (fun (x:Prop)=> (a x)))
% 21.56/21.76  Found (eta_expansion_dep00 a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.56/21.76  Found ((eta_expansion_dep0 (fun (x1:Prop)=> Prop)) a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.56/21.76  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.56/21.76  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.56/21.76  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.56/21.76  Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (Prop->Prop)) b0) (fun (x:Prop)=> (b0 x)))
% 21.56/21.76  Found (eta_expansion_dep00 b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.56/21.76  Found ((eta_expansion_dep0 (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.56/21.76  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.56/21.76  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.79/21.97  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.79/21.97  Found ((eq_trans00100 (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.79/21.97  Found ((eq_trans00100 (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.79/21.97  Found (((fun (x0:(((eq (Prop->Prop)) a) b0)) (x1:(((eq (Prop->Prop)) b0) c))=> (((eq_trans0010 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.79/21.97  Found (((fun (x0:(((eq (Prop->Prop)) a) b0)) (x1:(((eq (Prop->Prop)) b0) (fun (x2:Prop)=> (b0 x2))))=> ((((eq_trans001 (fun (x2:Prop)=> (b0 x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.79/21.97  Found (((fun (x0:(((eq (Prop->Prop)) a) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) (x1:(((eq (Prop->Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))))=> (((((eq_trans00 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.79/21.97  Found (((fun (x0:(((eq (Prop->Prop)) a) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) (x1:(((eq (Prop->Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))))=> (((((eq_trans00 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.79/21.97  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) (x1:(((eq (Prop->Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))))=> (((((eq_trans00 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.79/21.97  Found eta_expansion000:=(eta_expansion00 a):(((eq (Prop->Prop)) a) (fun (x:Prop)=> (a x)))
% 21.85/22.05  Found (eta_expansion00 a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.85/22.05  Found ((eta_expansion0 Prop) a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.85/22.05  Found (((eta_expansion Prop) Prop) a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.85/22.05  Found (((eta_expansion Prop) Prop) a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.85/22.05  Found (((eta_expansion Prop) Prop) a) as proof of (((eq (Prop->Prop)) a) b0)
% 21.85/22.05  Found eta_expansion_dep000:=(eta_expansion_dep00 b0):(((eq (Prop->Prop)) b0) (fun (x:Prop)=> (b0 x)))
% 21.85/22.05  Found (eta_expansion_dep00 b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.85/22.05  Found ((eta_expansion_dep0 (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.85/22.05  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.85/22.05  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.85/22.05  Found (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0) as proof of (((eq (Prop->Prop)) b0) c)
% 21.85/22.05  Found ((eq_trans00100 (((eta_expansion Prop) Prop) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.85/22.05  Found ((eq_trans00100 (((eta_expansion Prop) Prop) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.85/22.05  Found (((fun (x0:(((eq (Prop->Prop)) a) b0)) (x1:(((eq (Prop->Prop)) b0) c))=> (((eq_trans0010 x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.85/22.05  Found (((fun (x0:(((eq (Prop->Prop)) a) b0)) (x1:(((eq (Prop->Prop)) b0) (fun (x2:Prop)=> (b0 x2))))=> ((((eq_trans001 (fun (x2:Prop)=> (b0 x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) b0)) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.85/22.05  Found (((fun (x0:(((eq (Prop->Prop)) a) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) (x1:(((eq (Prop->Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))))=> (((((eq_trans00 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.85/22.05  Found (((fun (x0:(((eq (Prop->Prop)) a) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) (x1:(((eq (Prop->Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))))=> (((((eq_trans00 (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2))))) (fun (x2:Prop)=> ((fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))) x2))) x0) x1) (fun (x2:(Prop->Prop))=> (P0 (((eq Prop) x) Xy))))) (((eta_expansion Prop) Prop) a)) (((eta_expansion_dep Prop) (fun (x1:Prop)=> Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) as proof of ((P0 (((eq Prop) x) Xy))->(P0 (((eq Prop) Xy) x)))
% 21.85/22.05  Found (fun (P0:(Prop->Prop))=> (((fun (x0:(((eq (Prop->Prop)) a) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy0) x2)))))) (x1:(((eq (Prop->Prop)) (fun (x2:Prop)=> (forall (Xy0:Prop), (((eq Prop) (((eq Prop) x2) Xy0)) (((eq Prop) Xy
%------------------------------------------------------------------------------