TSTP Solution File: SEV109^5 by cocATP---0.2.0

%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEV109^5 : 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 13:33:45 EDT 2014

% Result   : Theorem 199.02s
% Output   : Proof 199.02s
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEV109^5 : 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 08:06:56 CDT 2014
% % CPUTime  : 199.02 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% FOF formula (<kernel.Constant object at 0x1e4ec68>, <kernel.Type object at 0x1e4eb48>) of role type named a_type
% Using role type
% Declaring a:Type
% FOF formula (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))))) of role conjecture named cTHM251F_pme
% Conjecture to prove = (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))))):Prop
% Parameter a_DUMMY:a.
% We need to prove ['(forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))))']
% Parameter a:Type.
% Trying to prove (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))))
% Found x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x2 as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x3:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x3 as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x5:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))
% Found x5 as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))
% Found x5:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))
% Found x5 as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))
% Found x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))
% Found x6 as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))
% Found x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))
% Found x6 as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))
% Found x5:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))
% Found x5 as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))
% Found x5:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))
% Found x5 as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))
% Found x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))
% Found x6 as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))
% Found x6:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))
% Found x6 as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))
% Found x500:=(x50 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x50 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x500:=(x50 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x50 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x400:=(x40 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x40 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x400:=(x40 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x40 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x500:=(x50 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x50 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x500:=(x50 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x50 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x400:=(x40 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x40 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x400:=(x40 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x40 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x500:=(x50 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x50 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x500:=(x50 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x50 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x40000:=(x4000 x2):((Xp1 Xx0) Xz)
% Found (x4000 x2) as proof of ((Xp1 Xx0) Xz)
% Found ((x400 Xy0) x2) as proof of ((Xp1 Xx0) Xz)
% Found (((fun (Xy00:a)=> ((x40 Xy00) Xz)) Xy0) x2) as proof of ((Xp1 Xx0) Xz)
% Found (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2) as proof of ((Xp1 Xx0) Xz)
% Found (fun (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)) as proof of ((Xp1 Xx0) Xz)
% Found (fun (x3:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq2 Xx00) Xy0)) ((Xq2 Xy0) Xz0))->((Xq2 Xx00) Xz0))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq3 Xx00) Xy0)) ((Xq3 Xy0) Xz0))->((Xq3 Xx00) Xz0))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)) as proof of ((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz))
% Found (fun (x3:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq2 Xx00) Xy0)) ((Xq2 Xy0) Xz0))->((Xq2 Xx00) Xz0))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq3 Xx00) Xy0)) ((Xq3 Xy0) Xz0))->((Xq3 Xx00) Xz0))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)) as proof of ((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq2 Xx00) Xy0)) ((Xq2 Xy0) Xz0))->((Xq2 Xx00) Xz0))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq3 Xx00) Xy0)) ((Xq3 Xy0) Xz0))->((Xq3 Xx00) Xz0))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0)))->((Xp1 Xx0) Xz)))
% Found (and_rect00 (fun (x3:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq2 Xx00) Xy0)) ((Xq2 Xy0) Xz0))->((Xq2 Xx00) Xz0))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq3 Xx00) Xy0)) ((Xq3 Xy0) Xz0))->((Xq3 Xx00) Xz0))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2))) as proof of ((Xp1 Xx0) Xz)
% Found ((and_rect0 ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq2 Xx00) Xy0)) ((Xq2 Xy0) Xz0))->((Xq2 Xx00) Xz0))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq3 Xx00) Xy0)) ((Xq3 Xy0) Xz0))->((Xq3 Xx00) Xz0))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2))) as proof of ((Xp1 Xx0) Xz)
% Found (((fun (P:Type) (x3:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x3) x0)) ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq2 Xx00) Xy0)) ((Xq2 Xy0) Xz0))->((Xq2 Xx00) Xz0))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq3 Xx00) Xy0)) ((Xq3 Xy0) Xz0))->((Xq3 Xx00) Xz0))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2))) as proof of ((Xp1 Xx0) Xz)
% Found (fun (x2:((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz)))=> (((fun (P:Type) (x3:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x3) x0)) ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq2 Xx00) Xy0)) ((Xq2 Xy0) Xz0))->((Xq2 Xx00) Xz0))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq3 Xx00) Xy0)) ((Xq3 Xy0) Xz0))->((Xq3 Xx00) Xz0))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)))) as proof of ((Xp1 Xx0) Xz)
% Found (fun (Xz:a) (x2:((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz)))=> (((fun (P:Type) (x3:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x3) x0)) ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq2 Xx00) Xy0)) ((Xq2 Xy0) Xz0))->((Xq2 Xx00) Xz0))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq3 Xx00) Xy0)) ((Xq3 Xy0) Xz0))->((Xq3 Xx00) Xz0))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)))) as proof of (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))
% Found (fun (Xy0:a) (Xz:a) (x2:((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz)))=> (((fun (P:Type) (x3:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x3) x0)) ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq2 Xx00) Xy0)) ((Xq2 Xy0) Xz0))->((Xq2 Xx00) Xz0))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq3 Xx00) Xy0)) ((Xq3 Xy0) Xz0))->((Xq3 Xx00) Xz0))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)))) as proof of (forall (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found (fun (Xx0:a) (Xy0:a) (Xz:a) (x2:((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz)))=> (((fun (P:Type) (x3:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x3) x0)) ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq2 Xx00) Xy0)) ((Xq2 Xy0) Xz0))->((Xq2 Xx00) Xz0))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq3 Xx00) Xy0)) ((Xq3 Xy0) Xz0))->((Xq3 Xx00) Xz0))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)))) as proof of (forall (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found (fun (Xx0:a) (Xy0:a) (Xz:a) (x2:((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz)))=> (((fun (P:Type) (x3:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x3) x0)) ((Xp1 Xx0) Xz)) (fun (x3:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq2 Xx00) Xy0)) ((Xq2 Xy0) Xz0))->((Xq2 Xx00) Xz0))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xq3 Xx00) Xy0)) ((Xq3 Xy0) Xz0))->((Xq3 Xx00) Xz0))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (x4:(forall (Xx00:a) (Xy0:a) (Xz0:a), (((and ((Xp1 Xx00) Xy0)) ((Xp1 Xy0) Xz0))->((Xp1 Xx00) Xz0))))=> (((fun (Xy00:a)=> (((x4 Xx0) Xy00) Xz)) Xy0) x2)))) as proof of (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x500:=(x50 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x50 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x500:=(x50 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x50 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x600:=(x60 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x60 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x600:=(x60 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x60 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x600:=(x60 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x60 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x600:=(x60 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x60 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x600:=(x60 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x60 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x600:=(x60 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x60 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x600:=(x60 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x60 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x600:=(x60 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x60 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x500:=(x50 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x50 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x500:=(x50 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x50 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x400:=(x40 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x40 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x400:=(x40 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x40 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x500:=(x50 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x50 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x500:=(x50 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x50 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x7:((R Xs) Xt)
% Found (fun (x7:((R Xs) Xt))=> x7) as proof of ((R Xs) Xt)
% Found (fun (x7:((R Xs) Xt))=> x7) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x7:((S Xs) Xt)
% Found (fun (x7:((S Xs) Xt))=> x7) as proof of ((S Xs) Xt)
% Found (fun (x7:((S Xs) Xt))=> x7) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x600:=(x60 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x60 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x600:=(x60 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x60 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x400:=(x40 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x40 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x400:=(x40 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x40 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x8:((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of ((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x8:((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of ((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x400:=(x40 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x40 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x400:=(x40 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x40 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x4000:=(x400 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x400 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x40 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x40 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x4000:=(x400 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x400 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x40 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x40 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x8:((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of ((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x8:((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of ((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x500:=(x50 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x50 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x500:=(x50 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x50 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x8:((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of ((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x8:((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of ((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x8:((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of ((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x8:((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of ((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x500:=(x50 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x50 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x500:=(x50 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x50 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x7:((R Xs) Xt)
% Found (fun (x7:((R Xs) Xt))=> x7) as proof of ((R Xs) Xt)
% Found (fun (x7:((R Xs) Xt))=> x7) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x7:((S Xs) Xt)
% Found (fun (x7:((S Xs) Xt))=> x7) as proof of ((S Xs) Xt)
% Found (fun (x7:((S Xs) Xt))=> x7) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x600:=(x60 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x60 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x600:=(x60 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x60 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x8:((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of ((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x8:((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of ((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x600:=(x60 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x60 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x600:=(x60 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x60 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x8:((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of ((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x8:((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of ((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x600:=(x60 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x60 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x600:=(x60 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x60 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x8:((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of ((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x8:((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of ((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x500:=(x50 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x50 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x500:=(x50 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x50 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x600:=(x60 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x60 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x600:=(x60 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x60 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x8:((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of ((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x8:((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of ((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x8:((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of ((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x8:((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of ((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x7:((S Xs) Xt)
% Found (fun (x7:((S Xs) Xt))=> x7) as proof of ((S Xs) Xt)
% Found (fun (x7:((S Xs) Xt))=> x7) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x500:=(x50 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x50 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x7:((R Xs) Xt)
% Found (fun (x7:((R Xs) Xt))=> x7) as proof of ((R Xs) Xt)
% Found (fun (x7:((R Xs) Xt))=> x7) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x500:=(x50 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x50 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x5 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x400:=(x40 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x40 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x400:=(x40 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x40 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x4 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x600:=(x60 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x60 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x600:=(x60 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x60 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x8:((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of ((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x8:((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of ((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x600:=(x60 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x60 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x600:=(x60 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x60 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x6 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x4000:=(x400 Xt):(((R Xs) Xt)->((Xq2 Xs) Xt))
% Found (x400 Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x40 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found ((x40 Xs) Xt) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x4000:=(x400 Xt):(((S Xs) Xt)->((Xq3 Xs) Xt))
% Found (x400 Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x40 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found ((x40 Xs) Xt) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x8:((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of ((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x8:((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of ((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x6000:=(x600 x5):((Xq3 Xs) Xt)
% Found (x600 x5) as proof of ((Xq3 Xs) Xt)
% Found ((x60 Xt) x5) as proof of ((Xq3 Xs) Xt)
% Found (((x6 Xs) Xt) x5) as proof of ((Xq3 Xs) Xt)
% Found (fun (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x5)) as proof of ((Xq3 Xs) Xt)
% Found (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x5)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->((Xq3 Xs) Xt))
% Found (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x5)) as proof of ((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->((Xq3 Xs) Xt)))
% Found (and_rect10 (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x5))) as proof of ((Xq3 Xs) Xt)
% Found ((and_rect1 ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x5))) as proof of ((Xq3 Xs) Xt)
% Found (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x4)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x5))) as proof of ((Xq3 Xs) Xt)
% Found (fun (x5:((S Xs) Xt))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x4)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x5)))) as proof of ((Xq3 Xs) Xt)
% Found (fun (x5:((S Xs) Xt))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x4)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x5)))) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x6000:=(x600 x5):((Xq2 Xs) Xt)
% Found (x600 x5) as proof of ((Xq2 Xs) Xt)
% Found ((x60 Xt) x5) as proof of ((Xq2 Xs) Xt)
% Found (((x6 Xs) Xt) x5) as proof of ((Xq2 Xs) Xt)
% Found (fun (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x5)) as proof of ((Xq2 Xs) Xt)
% Found (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x5)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->((Xq2 Xs) Xt))
% Found (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x5)) as proof of ((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->((Xq2 Xs) Xt)))
% Found (and_rect10 (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x5))) as proof of ((Xq2 Xs) Xt)
% Found ((and_rect1 ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x5))) as proof of ((Xq2 Xs) Xt)
% Found (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x4)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x5))) as proof of ((Xq2 Xs) Xt)
% Found (fun (x5:((R Xs) Xt))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x4)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x5)))) as proof of ((Xq2 Xs) Xt)
% Found (fun (x5:((R Xs) Xt))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x4)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x5)))) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x8:((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of ((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x8:((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of ((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x7000:=(x700 x6):((Xq3 Xs) Xt)
% Found (x700 x6) as proof of ((Xq3 Xs) Xt)
% Found ((x70 Xt) x6) as proof of ((Xq3 Xs) Xt)
% Found (((x7 Xs) Xt) x6) as proof of ((Xq3 Xs) Xt)
% Found (fun (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6)) as proof of ((Xq3 Xs) Xt)
% Found (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->((Xq3 Xs) Xt))
% Found (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6)) as proof of ((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->((Xq3 Xs) Xt)))
% Found (and_rect10 (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6))) as proof of ((Xq3 Xs) Xt)
% Found ((and_rect1 ((Xq3 Xs) Xt)) (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6))) as proof of ((Xq3 Xs) Xt)
% Found (((fun (P:Type) (x7:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x7) x5)) ((Xq3 Xs) Xt)) (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6))) as proof of ((Xq3 Xs) Xt)
% Found (fun (x6:((S Xs) Xt))=> (((fun (P:Type) (x7:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x7) x5)) ((Xq3 Xs) Xt)) (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6)))) as proof of ((Xq3 Xs) Xt)
% Found (fun (x6:((S Xs) Xt))=> (((fun (P:Type) (x7:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x7) x5)) ((Xq3 Xs) Xt)) (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6)))) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x7000:=(x700 x6):((Xq2 Xs) Xt)
% Found (x700 x6) as proof of ((Xq2 Xs) Xt)
% Found ((x70 Xt) x6) as proof of ((Xq2 Xs) Xt)
% Found (((x7 Xs) Xt) x6) as proof of ((Xq2 Xs) Xt)
% Found (fun (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6)) as proof of ((Xq2 Xs) Xt)
% Found (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->((Xq2 Xs) Xt))
% Found (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6)) as proof of ((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->((Xq2 Xs) Xt)))
% Found (and_rect10 (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6))) as proof of ((Xq2 Xs) Xt)
% Found ((and_rect1 ((Xq2 Xs) Xt)) (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6))) as proof of ((Xq2 Xs) Xt)
% Found (((fun (P:Type) (x7:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x7) x5)) ((Xq2 Xs) Xt)) (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6))) as proof of ((Xq2 Xs) Xt)
% Found (fun (x6:((R Xs) Xt))=> (((fun (P:Type) (x7:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x7) x5)) ((Xq2 Xs) Xt)) (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6)))) as proof of ((Xq2 Xs) Xt)
% Found (fun (x6:((R Xs) Xt))=> (((fun (P:Type) (x7:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x7) x5)) ((Xq2 Xs) Xt)) (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6)))) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x8:((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of ((S Xs) Xt)
% Found (fun (x8:((S Xs) Xt))=> x8) as proof of (((S Xs) Xt)->((S Xs) Xt))
% Found x8:((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of ((R Xs) Xt)
% Found (fun (x8:((R Xs) Xt))=> x8) as proof of (((R Xs) Xt)->((R Xs) Xt))
% Found x7000:=(x700 x6):((Xq3 Xs) Xt)
% Found (x700 x6) as proof of ((Xq3 Xs) Xt)
% Found ((x70 Xt) x6) as proof of ((Xq3 Xs) Xt)
% Found (((x7 Xs) Xt) x6) as proof of ((Xq3 Xs) Xt)
% Found (fun (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6)) as proof of ((Xq3 Xs) Xt)
% Found (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->((Xq3 Xs) Xt))
% Found (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6)) as proof of ((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->((Xq3 Xs) Xt)))
% Found (and_rect10 (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6))) as proof of ((Xq3 Xs) Xt)
% Found ((and_rect1 ((Xq3 Xs) Xt)) (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6))) as proof of ((Xq3 Xs) Xt)
% Found (((fun (P:Type) (x7:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x7) x5)) ((Xq3 Xs) Xt)) (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6))) as proof of ((Xq3 Xs) Xt)
% Found (fun (x6:((S Xs) Xt))=> (((fun (P:Type) (x7:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x7) x5)) ((Xq3 Xs) Xt)) (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6)))) as proof of ((Xq3 Xs) Xt)
% Found (fun (x6:((S Xs) Xt))=> (((fun (P:Type) (x7:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x7) x5)) ((Xq3 Xs) Xt)) (fun (x7:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x7 Xs) Xt) x6)))) as proof of (((S Xs) Xt)->((Xq3 Xs) Xt))
% Found x7000:=(x700 x6):((Xq2 Xs) Xt)
% Found (x700 x6) as proof of ((Xq2 Xs) Xt)
% Found ((x70 Xt) x6) as proof of ((Xq2 Xs) Xt)
% Found (((x7 Xs) Xt) x6) as proof of ((Xq2 Xs) Xt)
% Found (fun (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6)) as proof of ((Xq2 Xs) Xt)
% Found (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->((Xq2 Xs) Xt))
% Found (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6)) as proof of ((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->((Xq2 Xs) Xt)))
% Found (and_rect10 (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6))) as proof of ((Xq2 Xs) Xt)
% Found ((and_rect1 ((Xq2 Xs) Xt)) (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6))) as proof of ((Xq2 Xs) Xt)
% Found (((fun (P:Type) (x7:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x7) x5)) ((Xq2 Xs) Xt)) (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6))) as proof of ((Xq2 Xs) Xt)
% Found (fun (x6:((R Xs) Xt))=> (((fun (P:Type) (x7:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x7) x5)) ((Xq2 Xs) Xt)) (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6)))) as proof of ((Xq2 Xs) Xt)
% Found (fun (x6:((R Xs) Xt))=> (((fun (P:Type) (x7:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x7) x5)) ((Xq2 Xs) Xt)) (fun (x7:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x8:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x7 Xs) Xt) x6)))) as proof of (((R Xs) Xt)->((Xq2 Xs) Xt))
% Found x6000:=(x600 x4):((Xq2 Xs) Xt)
% Found (x600 x4) as proof of ((Xq2 Xs) Xt)
% Found ((x60 Xt) x4) as proof of ((Xq2 Xs) Xt)
% Found (((x6 Xs) Xt) x4) as proof of ((Xq2 Xs) Xt)
% Found (fun (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4)) as proof of ((Xq2 Xs) Xt)
% Found (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->((Xq2 Xs) Xt))
% Found (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4)) as proof of ((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->((Xq2 Xs) Xt)))
% Found (and_rect10 (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))) as proof of ((Xq2 Xs) Xt)
% Found ((and_rect1 ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))) as proof of ((Xq2 Xs) Xt)
% Found (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))) as proof of ((Xq2 Xs) Xt)
% Found (fun (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))) as proof of ((Xq2 Xs) Xt)
% Found (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))) as proof of (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt))
% Found (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))) as proof of (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))
% Found (or_introl00 (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))) as proof of ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))
% Found ((or_introl0 (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))) as proof of ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))
% Found (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))) as proof of ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))
% Found (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))) as proof of ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))
% Found (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))) as proof of (((R Xs) Xt)->((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))))
% Found x6000:=(x600 x4):((Xq3 Xs) Xt)
% Found (x600 x4) as proof of ((Xq3 Xs) Xt)
% Found ((x60 Xt) x4) as proof of ((Xq3 Xs) Xt)
% Found (((x6 Xs) Xt) x4) as proof of ((Xq3 Xs) Xt)
% Found (fun (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)) as proof of ((Xq3 Xs) Xt)
% Found (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->((Xq3 Xs) Xt))
% Found (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)) as proof of ((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->((Xq3 Xs) Xt)))
% Found (and_rect10 (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))) as proof of ((Xq3 Xs) Xt)
% Found ((and_rect1 ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))) as proof of ((Xq3 Xs) Xt)
% Found (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))) as proof of ((Xq3 Xs) Xt)
% Found (fun (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))) as proof of ((Xq3 Xs) Xt)
% Found (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))) as proof of (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))
% Found (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))) as proof of (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))
% Found (or_intror00 (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))) as proof of ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))
% Found ((or_intror0 (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))) as proof of ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))
% Found (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))) as proof of ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))
% Found (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))) as proof of ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))
% Found (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))) as proof of (((S Xs) Xt)->((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))))
% Found ((or_ind00 (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) as proof of ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))
% Found (((or_ind0 ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) as proof of ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))
% Found ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) as proof of ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))
% Found ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) as proof of ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))
% Found (x200 ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))) as proof of ((Xp1 Xs) Xt)
% Found ((x20 Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))) as proof of ((Xp1 Xs) Xt)
% Found (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))) as proof of ((Xp1 Xs) Xt)
% Found (fun (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))))) as proof of ((Xp1 Xs) Xt)
% Found (fun (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))))) as proof of (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt))
% Found (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))))) as proof of (forall (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))
% Found (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))))) as proof of (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))
% Found (conj00 (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))
% Found ((conj0 (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))
% Found (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))
% Found (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))))))) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))
% Found (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))))))) as proof of ((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))))
% Found (and_rect00 (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))))) as proof of ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found ((and_rect0 ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))))) as proof of ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))))) as proof of ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))))) as proof of ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found (x1 (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))))))))) as proof of ((Xp1 Xx) Xy)
% Found ((x Xp1) (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))))))))) as proof of ((Xp1 Xx) Xy)
% Found (fun (x0:((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> ((x Xp1) (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))))))) as proof of ((Xp1 Xx) Xy)
% Found (fun (Xp1:(a->(a->Prop))) (x0:((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> ((x Xp1) (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))))))) as proof of (((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))
% Found (fun (x:(forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> ((x Xp1) (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))))))) as proof of (forall (Xp1:(a->(a->Prop))), (((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))
% Found (fun (Xy:a) (x:(forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> ((x Xp1) (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))))))) as proof of ((forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))))
% Found (fun (Xx:a) (Xy:a) (x:(forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> ((x Xp1) (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))))))) as proof of (forall (Xy:a), ((forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))))
% Found (fun (S:(a->(a->Prop))) (Xx:a) (Xy:a) (x:(forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> ((x Xp1) (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))))))) as proof of (forall (Xx:a) (Xy:a), ((forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))))
% Found (fun (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a) (x:(forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> ((x Xp1) (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))))))) as proof of (forall (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))))
% Found (fun (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a) (x:(forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> ((x Xp1) (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4)))))))))))))) as proof of (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))))
% Got proof (fun (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a) (x:(forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> ((x Xp1) (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))))))))))
% Time elapsed = 197.441195s
% node=13677 cost=1532.000000 depth=46
% ::::::::::::::::::::::
% % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p
% % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% (fun (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a) (x:(forall (Xq1:(a->(a->Prop))), (((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xq1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq1 Xx0) Xy0)) ((Xq1 Xy0) Xz))->((Xq1 Xx0) Xz))))->((Xq1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> ((x Xp1) (((fun (P:Type) (x2:((forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x2) x0)) ((and (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))) (fun (x2:(forall (Xss:a) (Xtt:a), (((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xss) Xtt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xss) Xtt))))->((Xp1 Xss) Xtt))))=> (((conj (forall (Xs:a) (Xt:a), (((or ((R Xs) Xt)) ((S Xs) Xt))->((Xp1 Xs) Xt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xs:a) (Xt:a) (x3:((or ((R Xs) Xt)) ((S Xs) Xt)))=> (((x2 Xs) Xt) ((((fun (P:Prop) (x4:(((R Xs) Xt)->P)) (x5:(((S Xs) Xt)->P))=> ((((((or_ind ((R Xs) Xt)) ((S Xs) Xt)) P) x4) x5) x3)) ((or (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt))))) (fun (x4:((R Xs) Xt))=> (((or_introl (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq2:(a->(a->Prop))) (x5:((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz)))) P) x6) x5)) ((Xq2 Xs) Xt)) (fun (x6:(forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))) (fun (x4:((S Xs) Xt))=> (((or_intror (forall (Xq2:(a->(a->Prop))), (((and (forall (Xsss:a) (Xttt:a), (((R Xsss) Xttt)->((Xq2 Xsss) Xttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq2 Xx0) Xy0)) ((Xq2 Xy0) Xz))->((Xq2 Xx0) Xz))))->((Xq2 Xs) Xt)))) (forall (Xq3:(a->(a->Prop))), (((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))->((Xq3 Xs) Xt)))) (fun (Xq3:(a->(a->Prop))) (x5:((and (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))))=> (((fun (P:Type) (x6:((forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz)))) P) x6) x5)) ((Xq3 Xs) Xt)) (fun (x6:(forall (Xssss:a) (Xtttt:a), (((S Xssss) Xtttt)->((Xq3 Xssss) Xtttt)))) (x7:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xq3 Xx0) Xy0)) ((Xq3 Xy0) Xz))->((Xq3 Xx0) Xz))))=> (((x6 Xs) Xt) x4))))))))))))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------