%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV000^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n187.star.cs.uiowa.edu
% Model    : x86_64 x86_64
% CPU      : Intel(R) Xeon(R) CPU E5-2609 0 2.40GHz
% Memory   : 32286.75MB
% OS       : Linux 2.6.32-431.20.3.el6.x86_64
% CPULimit : 300s
% DateTime : Thu Jul 17 13:33:31 EDT 2014

% Result   : Timeout 300.02s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV000^5 : TPTP v6.1.0. Released v4.0.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n187.star.cs.uiowa.edu
% % Model    : x86_64 x86_64
% % CPU      : Intel(R) Xeon(R) CPU E5-2609 0 @ 2.40GHz
% % Memory   : 32286.75MB
% % OS       : Linux 2.6.32-431.20.3.el6.x86_64
% % CPULimit : 300
% % DateTime : Thu Jul 17 07:23:36 CDT 2014
% % CPUTime  : 300.02 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x23b88c0>, <kernel.Type object at 0x23b8ab8>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (JOIN:(a->(a->a))) (MEET:(a->(a->a))), (((and ((and ((and ((and ((and ((and ((and (forall (Xx:a), (((eq a) ((JOIN Xx) Xx)) Xx))) (forall (Xx:a), (((eq a) ((MEET Xx) Xx)) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN ((JOIN Xx) Xy)) Xz)) ((JOIN Xx) ((JOIN Xy) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET ((MEET Xx) Xy)) Xz)) ((MEET Xx) ((MEET Xy) Xz)))))) (forall (Xx:a) (Xy:a), (((eq a) ((JOIN Xx) Xy)) ((JOIN Xy) Xx))))) (forall (Xx:a) (Xy:a), (((eq a) ((MEET Xx) Xy)) ((MEET Xy) Xx))))) (forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))) (forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))->((iff (forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))))) of role conjecture named cMODULAR_EQUIV_THM_pme
% Conjecture to prove = (forall (JOIN:(a->(a->a))) (MEET:(a->(a->a))), (((and ((and ((and ((and ((and ((and ((and (forall (Xx:a), (((eq a) ((JOIN Xx) Xx)) Xx))) (forall (Xx:a), (((eq a) ((MEET Xx) Xx)) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN ((JOIN Xx) Xy)) Xz)) ((JOIN Xx) ((JOIN Xy) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET ((MEET Xx) Xy)) Xz)) ((MEET Xx) ((MEET Xy) Xz)))))) (forall (Xx:a) (Xy:a), (((eq a) ((JOIN Xx) Xy)) ((JOIN Xy) Xx))))) (forall (Xx:a) (Xy:a), (((eq a) ((MEET Xx) Xy)) ((MEET Xy) Xx))))) (forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))) (forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))->((iff (forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (JOIN:(a->(a->a))) (MEET:(a->(a->a))), (((and ((and ((and ((and ((and ((and ((and (forall (Xx:a), (((eq a) ((JOIN Xx) Xx)) Xx))) (forall (Xx:a), (((eq a) ((MEET Xx) Xx)) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN ((JOIN Xx) Xy)) Xz)) ((JOIN Xx) ((JOIN Xy) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET ((MEET Xx) Xy)) Xz)) ((MEET Xx) ((MEET Xy) Xz)))))) (forall (Xx:a) (Xy:a), (((eq a) ((JOIN Xx) Xy)) ((JOIN Xy) Xx))))) (forall (Xx:a) (Xy:a), (((eq a) ((MEET Xx) Xy)) ((MEET Xy) Xx))))) (forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))) (forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))->((iff (forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))))']
% Parameter a:Type.
% Trying to prove (forall (JOIN:(a->(a->a))) (MEET:(a->(a->a))), (((and ((and ((and ((and ((and ((and ((and (forall (Xx:a), (((eq a) ((JOIN Xx) Xx)) Xx))) (forall (Xx:a), (((eq a) ((MEET Xx) Xx)) Xx)))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN ((JOIN Xx) Xy)) Xz)) ((JOIN Xx) ((JOIN Xy) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET ((MEET Xx) Xy)) Xz)) ((MEET Xx) ((MEET Xy) Xz)))))) (forall (Xx:a) (Xy:a), (((eq a) ((JOIN Xx) Xy)) ((JOIN Xy) Xx))))) (forall (Xx:a) (Xy:a), (((eq a) ((MEET Xx) Xy)) ((MEET Xy) Xx))))) (forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))) (forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))->((iff (forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))))
% Found x0000:=(x000 Xz):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (x000 Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((x00 Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (((x0 Xx) Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (((x0 Xx) Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (x10 (((x0 Xx) Xy) Xz)) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz)) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found (fun (x1:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found (fun (Xz:a) (x1:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz))) as proof of ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))
% Found (fun (Xy:a) (Xz:a) (x1:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz))) as proof of (forall (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x1:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz))) as proof of (forall (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (x0:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))) (Xx:a) (Xy:a) (Xz:a) (x1:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz))) as proof of (forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (x0:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))) (Xx:a) (Xy:a) (Xz:a) (x1:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz))) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))->(forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x1:(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x1 as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found x2000:=(x200 Xz):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (x200 Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((x20 Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (((x2 Xx) Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (((x2 Xx) Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (x30 (((x2 Xx) Xy) Xz)) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz)) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found (fun (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found (fun (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))
% Found (fun (Xy:a) (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of (forall (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of (forall (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))) (Xx:a) (Xy:a) (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of (forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))) (Xx:a) (Xy:a) (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))->(forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x0000:=(x000 Xz):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (x000 Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((x00 Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (((x0 Xx) Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (((x0 Xx) Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (x10 (((x0 Xx) Xy) Xz)) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz)) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found (fun (x1:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found (fun (Xz:a) (x1:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz))) as proof of ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))
% Found (fun (Xy:a) (Xz:a) (x1:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz))) as proof of (forall (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x1:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz))) as proof of (forall (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (x0:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))) (Xx:a) (Xy:a) (Xz:a) (x1:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz))) as proof of (forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (x0:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))) (Xx:a) (Xy:a) (Xz:a) (x1:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x1 (fun (x3:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x3))) ((MEET ((JOIN Xx) Xy)) x3)))) (((x0 Xx) Xy) Xz))) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))->(forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x3:(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x3 as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x1:(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x1 as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (fun (x20:(forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))=> ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (fun (x20:(forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))=> ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))) as proof of (P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (fun (x20:(forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))=> ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (fun (x10:((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (x20:(forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))=> ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))) as proof of ((forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0))->(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (fun (x10:((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (x20:(forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))=> ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))) as proof of (P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x3:(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x3 as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))):(((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) ((JOIN Xx) ((JOIN Xx) Xz)))
% Found (eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))):(((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) ((JOIN Xx) ((JOIN Xx) Xz)))
% Found (eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))):(((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) ((JOIN Xx) ((JOIN Xx) Xz)))
% Found (eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))):(((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) ((JOIN Xx) ((JOIN Xx) Xz)))
% Found (eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found x1:(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x1 as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))):(((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) ((JOIN Xx) ((JOIN Xx) Xz)))
% Found (eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))):(((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) ((JOIN Xx) ((JOIN Xx) Xz)))
% Found (eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))):(((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) ((JOIN Xx) ((JOIN Xx) Xz)))
% Found (eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))):(((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) ((JOIN Xx) ((JOIN Xx) Xz)))
% Found (eq_ref0 ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((JOIN Xx) ((JOIN Xx) Xz))) as proof of (((eq a) ((JOIN Xx) ((JOIN Xx) Xz))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) Xz))
% Found x1:(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (fun (x30:(forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))=> x1) as proof of (P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (fun (x30:(forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))=> x1) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref00:=(eq_ref0 ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))):(((eq a) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (eq_ref0 ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) as proof of (((eq a) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) as proof of (((eq a) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) as proof of (((eq a) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) b)
% Found ((eq_ref a) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) as proof of (((eq a) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) b)
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x2000:=(x200 Xz):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (x200 Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((x20 Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (((x2 Xx) Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (((x2 Xx) Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (x30 (((x2 Xx) Xy) Xz)) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz)) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found (fun (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found (fun (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))
% Found (fun (Xy:a) (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of (forall (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of (forall (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))) (Xx:a) (Xy:a) (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of (forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))) (Xx:a) (Xy:a) (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))->(forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))))
% Found x4000:=(x400 Xz):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (x400 Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((x40 Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (((x4 Xx) Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (((x4 Xx) Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (x50 (((x4 Xx) Xy) Xz)) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found ((x5 (fun (x7:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x7))) ((MEET ((JOIN Xx) Xy)) x7)))) (((x4 Xx) Xy) Xz)) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found (fun (x5:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x5 (fun (x7:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x7))) ((MEET ((JOIN Xx) Xy)) x7)))) (((x4 Xx) Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found (fun (Xz:a) (x5:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x5 (fun (x7:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x7))) ((MEET ((JOIN Xx) Xy)) x7)))) (((x4 Xx) Xy) Xz))) as proof of ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))
% Found (fun (Xy:a) (Xz:a) (x5:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x5 (fun (x7:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x7))) ((MEET ((JOIN Xx) Xy)) x7)))) (((x4 Xx) Xy) Xz))) as proof of (forall (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x5:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x5 (fun (x7:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x7))) ((MEET ((JOIN Xx) Xy)) x7)))) (((x4 Xx) Xy) Xz))) as proof of (forall (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (x4:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))) (Xx:a) (Xy:a) (Xz:a) (x5:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x5 (fun (x7:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x7))) ((MEET ((JOIN Xx) Xy)) x7)))) (((x4 Xx) Xy) Xz))) as proof of (forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (x4:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))) (Xx:a) (Xy:a) (Xz:a) (x5:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x5 (fun (x7:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x7))) ((MEET ((JOIN Xx) Xy)) x7)))) (((x4 Xx) Xy) Xz))) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))->(forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x1:(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (fun (x30:(forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))=> x1) as proof of (P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (fun (x20:((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (x30:(forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))=> x1) as proof of ((forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (fun (x20:((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (x30:(forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))=> x1) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref00:=(eq_ref0 b):(((eq a) b) b)
% Found (eq_ref0 b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) b)
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x2000:=(x200 Xz):(((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (x200 Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found ((x20 Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (((x2 Xx) Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (((x2 Xx) Xy) Xz) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))
% Found (x30 (((x2 Xx) Xy) Xz)) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz)) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found (fun (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))
% Found (fun (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))
% Found (fun (Xy:a) (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of (forall (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (Xx:a) (Xy:a) (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of (forall (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))) (Xx:a) (Xy:a) (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of (forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))
% Found (fun (x2:(forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))) (Xx:a) (Xy:a) (Xz:a) (x3:(((eq a) ((JOIN Xx) Xz)) Xz))=> ((x3 (fun (x5:a)=> (((eq a) ((JOIN Xx) ((MEET Xy) x5))) ((MEET ((JOIN Xx) Xy)) x5)))) (((x2 Xx) Xy) Xz))) as proof of ((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))->(forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x3:(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found x3 as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))))
% Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Found eq_ref000:=(eq_ref0
% EOF
%------------------------------------------------------------------------------