%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : SEU996^5 : TPTP v6.1.0. Released v4.0.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n089.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 13:33:29 EDT 2014 % Result : Theorem 63.78s % Output : Proof 63.78s % Verified : % SZS Type : None (Parsing solution fails) % Syntax : Number of formulae : 0 % Comments : %------------------------------------------------------------------------------ %----ERROR: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % % Problem : SEU996^5 : TPTP v6.1.0. Released v4.0.0. % % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % % Computer : n089.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 11:49:06 CDT 2014 % % CPUTime : 63.78 % 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 0x267cb90>, <kernel.Type object at 0x267c368>) 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 ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz)))))->(forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))))) of role conjecture named cMODULAR_THM_pme % Conjecture to prove = (forall (JOIN:(a->(a->a))) (MEET:(a->(a->a))), (((and ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz)))))->(forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))))):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 ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz)))))->(forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))))'] % Parameter a:Type. % Trying to prove (forall (JOIN:(a->(a->a))) (MEET:(a->(a->a))), (((and ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz)))))->(forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))))) % Found x00:=(x0 (fun (x1:a)=> (P ((JOIN Xx) ((MEET Xy) Xz))))):((P ((JOIN Xx) ((MEET Xy) Xz)))->(P ((JOIN Xx) ((MEET Xy) Xz)))) % Found (x0 (fun (x1:a)=> (P ((JOIN Xx) ((MEET Xy) Xz))))) as proof of (P0 ((JOIN Xx) ((MEET Xy) Xz))) % Found (x0 (fun (x1:a)=> (P ((JOIN Xx) ((MEET Xy) Xz))))) as proof of (P0 ((JOIN Xx) ((MEET Xy) Xz))) % Found eq_ref000:=(eq_ref00 P):((P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))->(P ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz))))) % Found (eq_ref00 P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) % Found ((eq_ref0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) % Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) % Found (((eq_ref a) ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) P) as proof of (P0 ((JOIN Xx) ((MEET Xy) ((JOIN Xx) Xz)))) % Found eq_ref00:=(eq_ref0 ((JOIN Xx) ((MEET Xy) Xz))):(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((JOIN Xx) ((MEET Xy) Xz))) % Found (eq_ref0 ((JOIN Xx) ((MEET Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) b) % Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) b) % Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) b) % Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) 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)) Xz)) % Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) Xz)) % Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) Xz)) % Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) 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 x00:=(x0 (fun (x3:a)=> (P ((JOIN Xx) ((MEET Xy) Xz))))):((P ((JOIN Xx) ((MEET Xy) Xz)))->(P ((JOIN Xx) ((MEET Xy) Xz)))) % Found (x0 (fun (x3:a)=> (P ((JOIN Xx) ((MEET Xy) Xz))))) as proof of (P0 ((JOIN Xx) ((MEET Xy) Xz))) % Found (x0 (fun (x3:a)=> (P ((JOIN Xx) ((MEET Xy) Xz))))) as proof of (P0 ((JOIN Xx) ((MEET Xy) Xz))) % Found eq_ref000:=(eq_ref00 P):((P ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))->(P ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))) % Found (eq_ref00 P) as proof of (P0 ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) % Found ((eq_ref0 ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) P) as proof of (P0 ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) % Found (((eq_ref a) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) P) as proof of (P0 ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) % Found (((eq_ref a) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) P) as proof of (P0 ((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 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) Xz))):(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((JOIN Xx) ((MEET Xy) Xz))) % Found (eq_ref0 ((JOIN Xx) ((MEET Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) b) % Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) b) % Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) b) % Found ((eq_ref a) ((JOIN Xx) ((MEET Xy) Xz))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) 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)) Xz)) % Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) Xz)) % Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) Xz)) % Found ((eq_ref a) b) as proof of (((eq a) b) ((MEET ((JOIN Xx) Xy)) Xz)) % Found x400000:=(x40000 x5):(P ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) % Found (x40000 x5) as proof of (P ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) % Found ((x4000 P) x5) as proof of (P ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) % Found (((x400 Xz) P) x5) as proof of (P ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) % Found ((((x40 Xy) Xz) P) x5) as proof of (P ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) % Found (((((x4 Xx) Xy) Xz) P) x5) as proof of (P ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) % Found (((((x4 Xx) Xy) Xz) P) x5) as proof of (P ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz))) % Found (x00 (((((x4 Xx) Xy) Xz) P) x5)) as proof of (P ((MEET ((JOIN Xx) Xy)) Xz)) % Found ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5)) as proof of (P ((MEET ((JOIN Xx) Xy)) Xz)) % Found (fun (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))) as proof of (P ((MEET ((JOIN Xx) Xy)) Xz)) % Found (fun (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))) as proof of ((P ((JOIN Xx) ((MEET Xy) Xz)))->(P ((MEET ((JOIN Xx) Xy)) Xz))) % Found (fun (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)) % Found (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))) as proof of ((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) % Found (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))) as proof of (((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))) % Found (and_rect10 (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)) % Found ((and_rect1 (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)) % Found (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5)))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)) % Found (fun (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)) % Found (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))))) as proof of ((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0))))->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) % Found (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))))) as proof of (((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0))))->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))) % Found (and_rect00 (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5)))))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)) % Found ((and_rect0 (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5)))))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)) % Found (((fun (P:Type) (x1:(((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))->((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))->P)))=> (((((and_rect ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))) P) x1) x)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5)))))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)) % Found (fun (x0:(((eq a) ((JOIN Xx) Xz)) Xz))=> (((fun (P:Type) (x1:(((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))->((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))->P)))=> (((((and_rect ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))) P) x1) x)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))))))) as proof of (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)) % Found (fun (Xz:a) (x0:(((eq a) ((JOIN Xx) Xz)) Xz))=> (((fun (P:Type) (x1:(((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))->((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))->P)))=> (((((and_rect ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))) P) x1) x)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))))))) as proof of ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) % Found (fun (Xy:a) (Xz:a) (x0:(((eq a) ((JOIN Xx) Xz)) Xz))=> (((fun (P:Type) (x1:(((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))->((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))->P)))=> (((((and_rect ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))) P) x1) x)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))))))) as proof of (forall (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))) % Found (fun (Xx:a) (Xy:a) (Xz:a) (x0:(((eq a) ((JOIN Xx) Xz)) Xz))=> (((fun (P:Type) (x1:(((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))->((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))->P)))=> (((((and_rect ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))) P) x1) x)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))))))) as proof of (forall (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))) % Found (fun (x:((and ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz)))))) (Xx:a) (Xy:a) (Xz:a) (x0:(((eq a) ((JOIN Xx) Xz)) Xz))=> (((fun (P:Type) (x1:(((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))->((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))->P)))=> (((((and_rect ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))) P) x1) x)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))))))) as proof of (forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))) % Found (fun (MEET:(a->(a->a))) (x:((and ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz)))))) (Xx:a) (Xy:a) (Xz:a) (x0:(((eq a) ((JOIN Xx) Xz)) Xz))=> (((fun (P:Type) (x1:(((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))->((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))->P)))=> (((((and_rect ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))) P) x1) x)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))))))) as proof of (((and ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz)))))->(forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))))) % Found (fun (JOIN:(a->(a->a))) (MEET:(a->(a->a))) (x:((and ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz)))))) (Xx:a) (Xy:a) (Xz:a) (x0:(((eq a) ((JOIN Xx) Xz)) Xz))=> (((fun (P:Type) (x1:(((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))->((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))->P)))=> (((((and_rect ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))) P) x1) x)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))))))) as proof of (forall (MEET:(a->(a->a))), (((and ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz)))))->(forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))))) % Found (fun (JOIN:(a->(a->a))) (MEET:(a->(a->a))) (x:((and ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz)))))) (Xx:a) (Xy:a) (Xz:a) (x0:(((eq a) ((JOIN Xx) Xz)) Xz))=> (((fun (P:Type) (x1:(((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))->((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))->P)))=> (((((and_rect ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))) P) x1) x)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))))))) as proof of (forall (JOIN:(a->(a->a))) (MEET:(a->(a->a))), (((and ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz)))))->(forall (Xx:a) (Xy:a) (Xz:a), ((((eq a) ((JOIN Xx) Xz)) Xz)->(((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz)))))) % Got proof (fun (JOIN:(a->(a->a))) (MEET:(a->(a->a))) (x:((and ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz)))))) (Xx:a) (Xy:a) (Xz:a) (x0:(((eq a) ((JOIN Xx) Xz)) Xz))=> (((fun (P:Type) (x1:(((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))->((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))->P)))=> (((((and_rect ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))) P) x1) x)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))))))) % Time elapsed = 63.137786s % node=4236 cost=697.000000 depth=29 % :::::::::::::::::::::: % % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % (fun (JOIN:(a->(a->a))) (MEET:(a->(a->a))) (x:((and ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz)))))) (Xx:a) (Xy:a) (Xz:a) (x0:(((eq a) ((JOIN Xx) Xz)) Xz))=> (((fun (P:Type) (x1:(((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))->((forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))->P)))=> (((((and_rect ((and ((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) Xz))) ((MEET ((JOIN Xx) Xy)) ((JOIN Xx) Xz)))))) (forall (Xx:a) (Xy:a) (Xz:a), (((eq a) ((MEET Xx) ((JOIN Xy) Xz))) ((JOIN ((MEET Xx) Xy)) ((MEET Xx) Xz))))) P) x1) x)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x1:((and ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0)))))) (x2:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET Xx0) ((JOIN Xy0) Xz0))) ((JOIN ((MEET Xx0) Xy0)) ((MEET Xx0) Xz0)))))=> (((fun (P:Type) (x3:(((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))->P)))=> (((((and_rect ((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) P) x3) x1)) (((eq a) ((JOIN Xx) ((MEET Xy) Xz))) ((MEET ((JOIN Xx) Xy)) Xz))) (fun (x3:((and ((and ((and ((and ((and ((and ((and (forall (Xx0:a), (((eq a) ((JOIN Xx0) Xx0)) Xx0))) (forall (Xx0:a), (((eq a) ((MEET Xx0) Xx0)) Xx0)))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN ((JOIN Xx0) Xy0)) Xz0)) ((JOIN Xx0) ((JOIN Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((MEET ((MEET Xx0) Xy0)) Xz0)) ((MEET Xx0) ((MEET Xy0) Xz0)))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN Xx0) Xy0)) ((JOIN Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET Xx0) Xy0)) ((MEET Xy0) Xx0))))) (forall (Xx0:a) (Xy0:a), (((eq a) ((JOIN ((MEET Xx0) Xy0)) Xy0)) Xy0)))) (forall (Xx0:a) (Xy0:a), (((eq a) ((MEET ((JOIN Xx0) Xy0)) Xy0)) Xy0)))) (x4:(forall (Xx0:a) (Xy0:a) (Xz0:a), (((eq a) ((JOIN Xx0) ((MEET Xy0) Xz0))) ((MEET ((JOIN Xx0) Xy0)) ((JOIN Xx0) Xz0))))) (P:(a->Prop)) (x5:(P ((JOIN Xx) ((MEET Xy) Xz))))=> ((x0 (fun (x7:a)=> (P ((MEET ((JOIN Xx) Xy)) x7)))) (((((x4 Xx) Xy) Xz) P) x5))))))) % % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % EOF %------------------------------------------------------------------------------