TSTP Solution File: SEV003^5 by cocATP---0.2.0
View Problem
- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV003^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n091.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 300.11s
% Output : None
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV003^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n091.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 07:24:11 CDT 2014
% % CPUTime : 300.11
% 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 0x10a5950>, <kernel.Type object at 0x10a5908>) 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)))->(((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))->False)->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) of role conjecture named cPENTAGON_THM2C_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)))->(((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))->False)->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))):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)))->(((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))->False)->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))))']
% 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)))->(((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))))->False)->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))))
% Found eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) (fun (x:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x) Xy)))) (((eq a) ((MEET x) Xy)) Xy))) (((eq a) ((JOIN x) Xy)) x))) (((eq a) ((MEET x) Xa)) Xa))) (((eq a) ((JOIN x) Xa)) x))) (((eq a) ((MEET x) Xb)) Xb))) (((eq a) ((JOIN x) Xb)) x))) (((eq a) ((MEET x) Xc)) Xc))) (((eq a) ((JOIN x) Xc)) x))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x) Xy)))) (((eq a) ((MEET x) Xy)) Xy))) (((eq a) ((JOIN x) Xy)) x))) (((eq a) ((MEET x) Xa)) Xa))) (((eq a) ((JOIN x) Xa)) x))) (((eq a) ((MEET x) Xb)) Xb))) (((eq a) ((JOIN x) Xb)) x))) (((eq a) ((MEET x) Xc)) Xc))) (((eq a) ((JOIN x) Xc)) x))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x) Xy)))) (((eq a) ((MEET x) Xy)) Xy))) (((eq a) ((JOIN x) Xy)) x))) (((eq a) ((MEET x) Xa)) Xa))) (((eq a) ((JOIN x) Xa)) x))) (((eq a) ((MEET x) Xb)) Xb))) (((eq a) ((JOIN x) Xb)) x))) (((eq a) ((MEET x) Xc)) Xc))) (((eq a) ((JOIN x) Xc)) x))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found False_rect00:=(False_rect0 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))):((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (False_rect0 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x2:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x1:((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)))) (x2:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of ((forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy))->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))
% Found (fun (x1:((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)))) (x2:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of (((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))->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))
% Found (and_rect00 (fun (x1:((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)))) (x2:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found ((and_rect0 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x1:((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)))) (x2:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (((fun (P:Type) (x1:(((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))->P)))=> (((((and_rect ((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))) P) x1) x)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x1:((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)))) (x2:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (((fun (P:Type) (x1:(((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))->P)))=> (((((and_rect ((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))) P) x1) x)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x1:((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)))) (x2:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% 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)))) 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 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_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 eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) (fun (x:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x) Xy)))) (((eq a) ((MEET x) Xy)) Xy))) (((eq a) ((JOIN x) Xy)) x))) (((eq a) ((MEET x) Xa)) Xa))) (((eq a) ((JOIN x) Xa)) x))) (((eq a) ((MEET x) Xb)) Xb))) (((eq a) ((JOIN x) Xb)) x))) (((eq a) ((MEET x) Xc)) Xc))) (((eq a) ((JOIN x) Xc)) x))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found ((eta_expansion_dep0 (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found (((eta_expansion_dep a) (fun (x4:a)=> Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) 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 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 ((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 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 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x3)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x3)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x3)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x3) Xy)))) (((eq a) ((MEET x3) Xy)) Xy))) (((eq a) ((JOIN x3) Xy)) x3))) (((eq a) ((MEET x3) Xa)) Xa))) (((eq a) ((JOIN x3) Xa)) x3))) (((eq a) ((MEET x3) Xb)) Xb))) (((eq a) ((JOIN x3) Xb)) x3))) (((eq a) ((MEET x3) Xc)) Xc))) (((eq a) ((JOIN x3) Xc)) x3))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x3))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x3))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x3)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x3)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x3)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x3) Xy)))) (((eq a) ((MEET x3) Xy)) Xy))) (((eq a) ((JOIN x3) Xy)) x3))) (((eq a) ((MEET x3) Xa)) Xa))) (((eq a) ((JOIN x3) Xa)) x3))) (((eq a) ((MEET x3) Xb)) Xb))) (((eq a) ((JOIN x3) Xb)) x3))) (((eq a) ((MEET x3) Xc)) Xc))) (((eq a) ((JOIN x3) Xc)) x3))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x3))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x3))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x3)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x3)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x3)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x3) Xy)))) (((eq a) ((MEET x3) Xy)) Xy))) (((eq a) ((JOIN x3) Xy)) x3))) (((eq a) ((MEET x3) Xa)) Xa))) (((eq a) ((JOIN x3) Xa)) x3))) (((eq a) ((MEET x3) Xb)) Xb))) (((eq a) ((JOIN x3) Xb)) x3))) (((eq a) ((MEET x3) Xc)) Xc))) (((eq a) ((JOIN x3) Xc)) x3))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x3))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x3))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x3)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x3)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x3)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x3) Xy)))) (((eq a) ((MEET x3) Xy)) Xy))) (((eq a) ((JOIN x3) Xy)) x3))) (((eq a) ((MEET x3) Xa)) Xa))) (((eq a) ((JOIN x3) Xa)) x3))) (((eq a) ((MEET x3) Xb)) Xb))) (((eq a) ((JOIN x3) Xb)) x3))) (((eq a) ((MEET x3) Xc)) Xc))) (((eq a) ((JOIN x3) Xc)) x3))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x3))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x3))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x) Xy)))) (((eq a) ((MEET x) Xy)) Xy))) (((eq a) ((JOIN x) Xy)) x))) (((eq a) ((MEET x) Xa)) Xa))) (((eq a) ((JOIN x) Xa)) x))) (((eq a) ((MEET x) Xb)) Xb))) (((eq a) ((JOIN x) Xb)) x))) (((eq a) ((MEET x) Xc)) Xc))) (((eq a) ((JOIN x) Xc)) x))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found eq_ref00:=(eq_ref0 (f x3)):(((eq Prop) (f x3)) (f x3))
% Found (eq_ref0 (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x3)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x3)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x3)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x3) Xy)))) (((eq a) ((MEET x3) Xy)) Xy))) (((eq a) ((JOIN x3) Xy)) x3))) (((eq a) ((MEET x3) Xa)) Xa))) (((eq a) ((JOIN x3) Xa)) x3))) (((eq a) ((MEET x3) Xb)) Xb))) (((eq a) ((JOIN x3) Xb)) x3))) (((eq a) ((MEET x3) Xc)) Xc))) (((eq a) ((JOIN x3) Xc)) x3))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x3))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x3))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x3)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x3)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x3)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x3) Xy)))) (((eq a) ((MEET x3) Xy)) Xy))) (((eq a) ((JOIN x3) Xy)) x3))) (((eq a) ((MEET x3) Xa)) Xa))) (((eq a) ((JOIN x3) Xa)) x3))) (((eq a) ((MEET x3) Xb)) Xb))) (((eq a) ((JOIN x3) Xb)) x3))) (((eq a) ((MEET x3) Xc)) Xc))) (((eq a) ((JOIN x3) Xc)) x3))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x3))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x3))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x3)) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x3)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x3)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x3)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x3) Xy)))) (((eq a) ((MEET x3) Xy)) Xy))) (((eq a) ((JOIN x3) Xy)) x3))) (((eq a) ((MEET x3) Xa)) Xa))) (((eq a) ((JOIN x3) Xa)) x3))) (((eq a) ((MEET x3) Xb)) Xb))) (((eq a) ((JOIN x3) Xb)) x3))) (((eq a) ((MEET x3) Xc)) Xc))) (((eq a) ((JOIN x3) Xc)) x3))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x3))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x3))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (((eq Prop) (f x3)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x3)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x3)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x3)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x3) Xy)))) (((eq a) ((MEET x3) Xy)) Xy))) (((eq a) ((JOIN x3) Xy)) x3))) (((eq a) ((MEET x3) Xa)) Xa))) (((eq a) ((JOIN x3) Xa)) x3))) (((eq a) ((MEET x3) Xb)) Xb))) (((eq a) ((JOIN x3) Xb)) x3))) (((eq a) ((MEET x3) Xc)) Xc))) (((eq a) ((JOIN x3) Xc)) x3))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x3))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x3))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x3:a)=> ((eq_ref Prop) (f x3))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x) Xy)))) (((eq a) ((MEET x) Xy)) Xy))) (((eq a) ((JOIN x) Xy)) x))) (((eq a) ((MEET x) Xa)) Xa))) (((eq a) ((JOIN x) Xa)) x))) (((eq a) ((MEET x) Xb)) Xb))) (((eq a) ((JOIN x) Xb)) x))) (((eq a) ((MEET x) Xc)) Xc))) (((eq a) ((JOIN x) Xc)) x))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found False_rect00:=(False_rect0 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))):((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (False_rect0 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of ((forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy))->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))
% Found (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of (((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))->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))
% Found (and_rect10 (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found ((and_rect1 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (((fun (P:Type) (x3:(((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))->P)))=> (((((and_rect ((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))) P) x3) x1)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (((fun (P:Type) (x3:(((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))->P)))=> (((((and_rect ((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))) P) x3) x1)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% 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)))) 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 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 (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) 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 False_rect00:=(False_rect0 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))):((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (False_rect0 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of ((forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy))->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))
% Found (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of (((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))->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))
% Found (and_rect10 (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found ((and_rect1 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (((fun (P:Type) (x3:(((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))->P)))=> (((((and_rect ((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))) P) x3) x1)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x2:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> (((fun (P:Type) (x3:(((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))->P)))=> (((((and_rect ((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))) P) x3) x1)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x2:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> (((fun (P:Type) (x3:(((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))->P)))=> (((((and_rect ((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))) P) x3) x1)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))))) as proof of ((forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy))->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))
% 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)))) 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 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_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 False_rect00:=(False_rect0 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))):((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (False_rect0 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of ((forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy))->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))
% Found (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of (((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))->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))
% Found (and_rect10 (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found ((and_rect1 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (((fun (P:Type) (x3:(((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))->P)))=> (((((and_rect ((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))) P) x3) x1)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x2:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> (((fun (P:Type) (x3:(((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))->P)))=> (((((and_rect ((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))) P) x3) x1)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x1:((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)))) (x2:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> (((fun (P:Type) (x3:(((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))->P)))=> (((((and_rect ((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))) P) x3) x1)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))))) as proof of ((forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy))->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))
% Found (fun (x1:((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)))) (x2:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> (((fun (P:Type) (x3:(((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))->P)))=> (((((and_rect ((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))) P) x3) x1)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x3:((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))))) (x4:(forall (Xx:a) (Xy:a), (((eq a) ((JOIN ((MEET Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))))) as proof of (((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))->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))
% 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)))) 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 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 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 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 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 ((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 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 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 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 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x5)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x5)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x5)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x5) Xy)))) (((eq a) ((MEET x5) Xy)) Xy))) (((eq a) ((JOIN x5) Xy)) x5))) (((eq a) ((MEET x5) Xa)) Xa))) (((eq a) ((JOIN x5) Xa)) x5))) (((eq a) ((MEET x5) Xb)) Xb))) (((eq a) ((JOIN x5) Xb)) x5))) (((eq a) ((MEET x5) Xc)) Xc))) (((eq a) ((JOIN x5) Xc)) x5))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x5))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x5))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x5)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x5)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x5)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x5) Xy)))) (((eq a) ((MEET x5) Xy)) Xy))) (((eq a) ((JOIN x5) Xy)) x5))) (((eq a) ((MEET x5) Xa)) Xa))) (((eq a) ((JOIN x5) Xa)) x5))) (((eq a) ((MEET x5) Xb)) Xb))) (((eq a) ((JOIN x5) Xb)) x5))) (((eq a) ((MEET x5) Xc)) Xc))) (((eq a) ((JOIN x5) Xc)) x5))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x5))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x5))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x5)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x5)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x5)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x5) Xy)))) (((eq a) ((MEET x5) Xy)) Xy))) (((eq a) ((JOIN x5) Xy)) x5))) (((eq a) ((MEET x5) Xa)) Xa))) (((eq a) ((JOIN x5) Xa)) x5))) (((eq a) ((MEET x5) Xb)) Xb))) (((eq a) ((JOIN x5) Xb)) x5))) (((eq a) ((MEET x5) Xc)) Xc))) (((eq a) ((JOIN x5) Xc)) x5))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x5))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x5))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x5)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x5)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x5)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x5) Xy)))) (((eq a) ((MEET x5) Xy)) Xy))) (((eq a) ((JOIN x5) Xy)) x5))) (((eq a) ((MEET x5) Xa)) Xa))) (((eq a) ((JOIN x5) Xa)) x5))) (((eq a) ((MEET x5) Xb)) Xb))) (((eq a) ((JOIN x5) Xb)) x5))) (((eq a) ((MEET x5) Xc)) Xc))) (((eq a) ((JOIN x5) Xc)) x5))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x5))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x5))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x) Xy)))) (((eq a) ((MEET x) Xy)) Xy))) (((eq a) ((JOIN x) Xy)) x))) (((eq a) ((MEET x) Xa)) Xa))) (((eq a) ((JOIN x) Xa)) x))) (((eq a) ((MEET x) Xb)) Xb))) (((eq a) ((JOIN x) Xb)) x))) (((eq a) ((MEET x) Xc)) Xc))) (((eq a) ((JOIN x) Xc)) x))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found eq_ref00:=(eq_ref0 (f x5)):(((eq Prop) (f x5)) (f x5))
% Found (eq_ref0 (f x5)) as proof of (((eq Prop) (f x5)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x5)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x5)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x5)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x5) Xy)))) (((eq a) ((MEET x5) Xy)) Xy))) (((eq a) ((JOIN x5) Xy)) x5))) (((eq a) ((MEET x5) Xa)) Xa))) (((eq a) ((JOIN x5) Xa)) x5))) (((eq a) ((MEET x5) Xb)) Xb))) (((eq a) ((JOIN x5) Xb)) x5))) (((eq a) ((MEET x5) Xc)) Xc))) (((eq a) ((JOIN x5) Xc)) x5))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x5))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x5))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x5)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x5)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x5)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x5) Xy)))) (((eq a) ((MEET x5) Xy)) Xy))) (((eq a) ((JOIN x5) Xy)) x5))) (((eq a) ((MEET x5) Xa)) Xa))) (((eq a) ((JOIN x5) Xa)) x5))) (((eq a) ((MEET x5) Xb)) Xb))) (((eq a) ((JOIN x5) Xb)) x5))) (((eq a) ((MEET x5) Xc)) Xc))) (((eq a) ((JOIN x5) Xc)) x5))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x5))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x5))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x5)) as proof of (((eq Prop) (f x5)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x5)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x5)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x5)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x5) Xy)))) (((eq a) ((MEET x5) Xy)) Xy))) (((eq a) ((JOIN x5) Xy)) x5))) (((eq a) ((MEET x5) Xa)) Xa))) (((eq a) ((JOIN x5) Xa)) x5))) (((eq a) ((MEET x5) Xb)) Xb))) (((eq a) ((JOIN x5) Xb)) x5))) (((eq a) ((MEET x5) Xc)) Xc))) (((eq a) ((JOIN x5) Xc)) x5))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x5))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x5))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (((eq Prop) (f x5)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x5)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x5)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x5)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x5) Xy)))) (((eq a) ((MEET x5) Xy)) Xy))) (((eq a) ((JOIN x5) Xy)) x5))) (((eq a) ((MEET x5) Xa)) Xa))) (((eq a) ((JOIN x5) Xa)) x5))) (((eq a) ((MEET x5) Xb)) Xb))) (((eq a) ((JOIN x5) Xb)) x5))) (((eq a) ((MEET x5) Xc)) Xc))) (((eq a) ((JOIN x5) Xc)) x5))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x5))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x5))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x5:a)=> ((eq_ref Prop) (f x5))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x) Xy)))) (((eq a) ((MEET x) Xy)) Xy))) (((eq a) ((JOIN x) Xy)) x))) (((eq a) ((MEET x) Xa)) Xa))) (((eq a) ((JOIN x) Xa)) x))) (((eq a) ((MEET x) Xb)) Xb))) (((eq a) ((JOIN x) Xb)) x))) (((eq a) ((MEET x) Xc)) Xc))) (((eq a) ((JOIN x) Xc)) x))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% 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 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 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 ((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 eta_expansion_dep000:=(eta_expansion_dep00 (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) (fun (x:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x) Xy)))) (((eq a) ((MEET x) Xy)) Xy))) (((eq a) ((JOIN x) Xy)) x))) (((eq a) ((MEET x) Xa)) Xa))) (((eq a) ((JOIN x) Xa)) x))) (((eq a) ((MEET x) Xb)) Xb))) (((eq a) ((JOIN x) Xb)) x))) (((eq a) ((MEET x) Xc)) Xc))) (((eq a) ((JOIN x) Xc)) x))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (eta_expansion_dep00 (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found ((eta_expansion_dep0 (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found (((eta_expansion_dep a) (fun (x2:a)=> Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found False_rect00:=(False_rect0 ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))):((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))
% Found (False_rect0 ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))) as proof of ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))
% Found ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))) as proof of ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))
% Found (fun (x3:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))
% Found (fun (x2:((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)))) (x3:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of ((forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy))->((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x2:((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)))) (x3:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((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))->((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (and_rect00 (fun (x2:((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)))) (x3:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))
% Found ((and_rect0 ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))) (fun (x2:((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)))) (x3:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))
% Found (((fun (P:Type) (x2:(((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))->P)))=> (((((and_rect ((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))) P) x2) x)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))) (fun (x2:((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)))) (x3:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))
% Found (((fun (P:Type) (x2:(((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))->P)))=> (((((and_rect ((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))) P) x2) x)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))) (fun (x2:((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)))) (x3:(forall (Xx:a) (Xy:a), (((eq a) ((MEET ((JOIN Xx) Xy)) Xy)) Xy)))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))
% 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 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 x1:(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))
% Instantiate: b:=((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))):a
% Found x1 as proof of (P0 b)
% 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_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 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x) Xy)))) (((eq a) ((MEET x) Xy)) Xy))) (((eq a) ((JOIN x) Xy)) x))) (((eq a) ((MEET x) Xa)) Xa))) (((eq a) ((JOIN x) Xa)) x))) (((eq a) ((MEET x) Xb)) Xb))) (((eq a) ((JOIN x) Xb)) x))) (((eq a) ((MEET x) Xc)) Xc))) (((eq a) ((JOIN x) Xc)) x))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found eq_ref00:=(eq_ref0 (f x1)):(((eq Prop) (f x1)) (f x1))
% Found (eq_ref0 (f x1)) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found ((eq_ref Prop) (f x1)) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (((eq Prop) (f x1)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x1)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x1)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x1)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x1) Xy)))) (((eq a) ((MEET x1) Xy)) Xy))) (((eq a) ((JOIN x1) Xy)) x1))) (((eq a) ((MEET x1) Xa)) Xa))) (((eq a) ((JOIN x1) Xa)) x1))) (((eq a) ((MEET x1) Xb)) Xb))) (((eq a) ((JOIN x1) Xb)) x1))) (((eq a) ((MEET x1) Xc)) Xc))) (((eq a) ((JOIN x1) Xc)) x1))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x1))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x1))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))
% Found (fun (x1:a)=> ((eq_ref Prop) (f x1))) as proof of (forall (x:a), (((eq Prop) (f x)) ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) x)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) x)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) x)))) (not (((eq a) Xc) Xy)))) (not (((eq a) x) Xy)))) (((eq a) ((MEET x) Xy)) Xy))) (((eq a) ((JOIN x) Xy)) x))) (((eq a) ((MEET x) Xa)) Xa))) (((eq a) ((JOIN x) Xa)) x))) (((eq a) ((MEET x) Xb)) Xb))) (((eq a) ((JOIN x) Xb)) x))) (((eq a) ((MEET x) Xc)) Xc))) (((eq a) ((JOIN x) Xc)) x))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) x))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) x))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found eq_ref00:=(eq_ref0 (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))):(((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (eq_ref0 (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found ((eq_ref (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) as proof of (((eq (a->Prop)) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))) b)
% Found False_rect00:=(False_rect0 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))):((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (False_rect0 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x6:(forall (Xx:a) (Xy:a), (((eq a) ((MEET Xx) Xy)) ((MEET Xy) Xx))))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (fun (x5:((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))))) (x6:(forall (Xx:a) (Xy:a), (((eq a) ((MEET Xx) Xy)) ((MEET Xy) Xx))))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of ((forall (Xx:a) (Xy:a), (((eq a) ((MEET Xx) Xy)) ((MEET Xy) Xx)))->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))
% Found (fun (x5:((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))))) (x6:(forall (Xx:a) (Xy:a), (((eq a) ((MEET Xx) Xy)) ((MEET Xy) Xx))))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc)))))))))))))) as proof of (((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)))->((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))
% Found (and_rect20 (fun (x5:((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))))) (x6:(forall (Xx:a) (Xy:a), (((eq a) ((MEET Xx) Xy)) ((MEET Xy) Xx))))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found ((and_rect2 ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x5:((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))))) (x6:(forall (Xx:a) (Xy:a), (((eq a) ((MEET Xx) Xy)) ((MEET Xy) Xx))))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (((fun (P:Type) (x5:(((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)))->P)))=> (((((and_rect ((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)))) P) x5) x3)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x5:((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))))) (x6:(forall (Xx:a) (Xy:a), (((eq a) ((MEET Xx) Xy)) ((MEET Xy) Xx))))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% Found (((fun (P:Type) (x5:(((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)))->P)))=> (((((and_rect ((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)))) P) x5) x3)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))) (fun (x5:((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))))) (x6:(forall (Xx:a) (Xy:a), (((eq a) ((MEET Xx) Xy)) ((MEET Xy) Xx))))=> ((fun (P:Type)=> ((False_rect P) x00)) ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))))) as proof of ((ex a) (fun (Xx:a)=> ((ex a) (fun (Xy:a)=> ((ex a) (fun (Xa:a)=> ((ex a) (fun (Xb:a)=> ((ex a) (fun (Xc:a)=> ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and ((and (not (((eq a) Xa) Xb))) (not (((eq a) Xa) Xc)))) (not (((eq a) Xa) Xx)))) (not (((eq a) Xa) Xy)))) (not (((eq a) Xb) Xc)))) (not (((eq a) Xb) Xx)))) (not (((eq a) Xb) Xy)))) (not (((eq a) Xc) Xx)))) (not (((eq a) Xc) Xy)))) (not (((eq a) Xx) Xy)))) (((eq a) ((MEET Xx) Xy)) Xy))) (((eq a) ((JOIN Xx) Xy)) Xx))) (((eq a) ((MEET Xx) Xa)) Xa))) (((eq a) ((JOIN Xx) Xa)) Xx))) (((eq a) ((MEET Xx) Xb)) Xb))) (((eq a) ((JOIN Xx) Xb)) Xx))) (((eq a) ((MEET Xx) Xc)) Xc))) (((eq a) ((JOIN Xx) Xc)) Xx))) (((eq a) ((MEET Xa) Xb)) Xy))) (((eq a) ((JOIN Xa) Xb)) Xx))) (((eq a) ((MEET Xa) Xc)) Xa))) (((eq a) ((JOIN Xa) Xc)) Xc))) (((eq a) ((MEET Xa) Xy)) Xy))) (((eq a) ((JOIN Xa) Xy)) Xa))) (((eq a) ((MEET Xb) Xc)) Xy))) (((eq a) ((JOIN Xb) Xc)) Xx))) (((eq a) ((MEET Xb) Xy)) Xy))) (((eq a) ((JOIN Xb) Xy)) Xb))) (((eq a) ((MEET Xc) Xy)) Xy))) (((eq a) ((JOIN Xc) Xy)) Xc))))))))))))
% 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)))) 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 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)) ((
% EOF
%------------------------------------------------------------------------------