TSTP Solution File: SEV140^5 by cocATP---0.2.0
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- Process Solution
%------------------------------------------------------------------------------
% File : cocATP---0.2.0
% Problem : SEV140^5 : TPTP v6.1.0. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% Computer : n100.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 13:33:47 EDT 2014
% Result : Theorem 124.86s
% Output : Proof 124.86s
% Verified :
% SZS Type : None (Parsing solution fails)
% Syntax : Number of formulae : 0
% Comments :
%------------------------------------------------------------------------------
%----ERROR: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem : SEV140^5 : TPTP v6.1.0. Released v4.0.0.
% % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n100.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 08:12:56 CDT 2014
% % CPUTime : 124.86
% 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 0x1f65bd8>, <kernel.Type object at 0x1f64a70>) 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 (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))))) of role conjecture named cTHM250C_pme
% Conjecture to prove = (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x2 as proof of (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% 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 (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) 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 (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) 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 (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% 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 (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% 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 (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) 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 (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) 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 (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% 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 (Xx00:a) (Xy0:a), (((or ((R Xx00) Xy0)) ((S Xx00) Xy0))->((Xp1 Xx00) Xy0)))) (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 (Xx00:a) (Xy0:a), (((or ((R Xx00) Xy0)) ((S Xx00) Xy0))->((Xp1 Xx00) Xy0)))) (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), (((or ((R Xx00) Xy0)) ((S Xx00) Xy0))->((Xp1 Xx00) Xy0)))->((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 (Xx00:a) (Xy0:a), (((or ((R Xx00) Xy0)) ((S Xx00) Xy0))->((Xp1 Xx00) Xy0)))) (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 (Xx00:a) (Xy0:a), (((or ((R Xx00) Xy0)) ((S Xx00) Xy0))->((Xp1 Xx00) Xy0)))) (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 (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xx00:a) (Xy0:a), (((or ((R Xx00) Xy0)) ((S Xx00) Xy0))->((Xp1 Xx00) Xy0)))) (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 (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xx00:a) (Xy0:a), (((or ((R Xx00) Xy0)) ((S Xx00) Xy0))->((Xp1 Xx00) Xy0)))) (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 (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xx00:a) (Xy0:a), (((or ((R Xx00) Xy0)) ((S Xx00) Xy0))->((Xp1 Xx00) Xy0)))) (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 (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xx00:a) (Xy0:a), (((or ((R Xx00) Xy0)) ((S Xx00) Xy0))->((Xp1 Xx00) Xy0)))) (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 (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xx00:a) (Xy0:a), (((or ((R Xx00) Xy0)) ((S Xx00) Xy0))->((Xp1 Xx00) Xy0)))) (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 (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xx00:a) (Xy0:a), (((or ((R Xx00) Xy0)) ((S Xx00) Xy0))->((Xp1 Xx00) Xy0)))) (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 x4:(forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))
% Found x4 as proof of (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% Found x4:(forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))
% Found x4 as proof of (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% Found x5:(forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))
% Found x5 as proof of (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% Found x5:(forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))
% Found x5 as proof of (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% Found x5:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x5 as proof of (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% Found x5:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))
% Found x5 as proof of (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% Found or_intror00:=(or_intror0 ((S Xx1) Xy1)):(((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_intror0 ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_introl00:=(or_introl0 ((S Xx1) Xy1)):(((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_introl0 ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_introl00:=(or_introl0 ((S Xx1) Xy1)):(((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_introl0 ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_intror00:=(or_intror0 ((S Xx1) Xy1)):(((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_intror0 ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_introl00:=(or_introl0 ((S Xx1) Xy1)):(((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_introl0 ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_intror00:=(or_intror0 ((S Xx1) Xy1)):(((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_intror0 ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_intror00:=(or_intror0 ((S Xx1) Xy1)):(((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_intror0 ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_introl00:=(or_introl0 ((S Xx1) Xy1)):(((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_introl0 ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_intror00:=(or_intror0 ((S Xx1) Xy1)):(((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_intror0 ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_introl00:=(or_introl0 ((S Xx1) Xy1)):(((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_introl0 ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_intror00:=(or_intror0 ((S Xx1) Xy1)):(((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_intror0 ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_introl00:=(or_introl0 ((S Xx1) Xy1)):(((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_introl0 ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_intror00:=(or_intror0 ((S Xx1) Xy1)):(((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_intror0 ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_introl00:=(or_introl0 ((S Xx1) Xy1)):(((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_introl0 ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_intror00:=(or_intror0 ((S Xx1) Xy1)):(((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_intror0 ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_introl00:=(or_introl0 ((S Xx1) Xy1)):(((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_introl0 ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found x20:(forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))
% Found x20 as proof of (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% Found x20:(forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))
% Found x20 as proof of (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% Found or_introl00:=(or_introl0 ((S Xx1) Xy1)):(((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_introl0 ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_intror00:=(or_intror0 ((S Xx1) Xy1)):(((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_intror0 ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found x200:(forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))
% Found x200 as proof of (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% Found x200:(forall (Xx00:a) (Xy00:a) (Xz:a), (((and ((Xp1 Xx00) Xy00)) ((Xp1 Xy00) Xz))->((Xp1 Xx00) Xz)))
% Found x200 as proof of (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))
% Found or_introl00:=(or_introl0 ((S Xx1) Xy1)):(((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_introl0 ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_intror00:=(or_intror0 ((S Xx1) Xy1)):(((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_intror0 ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_intror00:=(or_intror0 ((S Xx1) Xy1)):(((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_intror0 ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_introl00:=(or_introl0 ((S Xx1) Xy1)):(((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_introl0 ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_introl00:=(or_introl0 ((S Xx1) Xy1)):(((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_introl0 ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_introl ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_intror00:=(or_intror0 ((S Xx1) Xy1)):(((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (or_intror0 ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found (fun (Xx1:a) (Xy1:a)=> ((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1))) as proof of (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((or ((R Xx1) Xy1)) ((S Xx1) Xy1))))
% Found or_introl000:=(or_introl00 ((S Xx1) Xy1)):((or ((R Xx1) Xy1)) ((S Xx1) Xy1))
% Found (or_introl00 ((S Xx1) Xy1)) as proof of ((or ((R Xx1) Xy1)) ((S Xx1) Xy1))
% Found ((fun (B:Prop)=> ((or_introl0 B) x6)) ((S Xx1) Xy1)) as proof of ((or ((R Xx1) Xy1)) ((S Xx1) Xy1))
% Found ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1)) as proof of ((or ((R Xx1) Xy1)) ((S Xx1) Xy1))
% Found ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1)) as proof of ((or ((R Xx1) Xy1)) ((S Xx1) Xy1))
% Found (x100 ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))) as proof of ((Xp1 Xx1) Xy1)
% Found ((x10 Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))) as proof of ((Xp1 Xx1) Xy1)
% Found (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))) as proof of ((Xp1 Xx1) Xy1)
% Found (fun (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1)))) as proof of ((Xp1 Xx1) Xy1)
% Found (fun (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1)))) as proof of (((R Xx1) Xy1)->((Xp1 Xx1) Xy1))
% Found (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1)))) as proof of (forall (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1)))) as proof of (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))
% Found ((conj10 (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2) as proof of ((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))
% Found (((conj1 (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2) as proof of ((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))
% Found ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2) as proof of ((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))
% Found ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2) as proof of ((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))
% Found (x50 ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)) as proof of ((Xp1 Xx0) Xy0)
% Found ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)) as proof of ((Xp1 Xx0) Xy0)
% Found (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2))) as proof of ((Xp1 Xx0) Xy0)
% Found (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2))) as proof of ((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->((Xp1 Xx0) Xy0))
% Found or_intror000:=(or_intror00 x6):((or ((R Xx1) Xy1)) ((S Xx1) Xy1))
% Found (or_intror00 x6) as proof of ((or ((R Xx1) Xy1)) ((S Xx1) Xy1))
% Found ((or_intror0 ((S Xx1) Xy1)) x6) as proof of ((or ((R Xx1) Xy1)) ((S Xx1) Xy1))
% Found (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6) as proof of ((or ((R Xx1) Xy1)) ((S Xx1) Xy1))
% Found (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6) as proof of ((or ((R Xx1) Xy1)) ((S Xx1) Xy1))
% Found (x100 (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)) as proof of ((Xp1 Xx1) Xy1)
% Found ((x10 Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)) as proof of ((Xp1 Xx1) Xy1)
% Found (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)) as proof of ((Xp1 Xx1) Xy1)
% Found (fun (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6))) as proof of ((Xp1 Xx1) Xy1)
% Found (fun (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6))) as proof of (((S Xx1) Xy1)->((Xp1 Xx1) Xy1))
% Found (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6))) as proof of (forall (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))
% Found (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6))) as proof of (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))
% Found ((conj10 (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2) as proof of ((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))
% Found (((conj1 (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2) as proof of ((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))
% Found ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2) as proof of ((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))
% Found ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2) as proof of ((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz))))
% Found (x50 ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)) as proof of ((Xp1 Xx0) Xy0)
% Found ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)) as proof of ((Xp1 Xx0) Xy0)
% Found (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2))) as proof of ((Xp1 Xx0) Xy0)
% Found (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2))) as proof of ((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->((Xp1 Xx0) Xy0))
% Found ((or_ind00 (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))) as proof of ((Xp1 Xx0) Xy0)
% Found (((or_ind0 ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))) as proof of ((Xp1 Xx0) Xy0)
% Found ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))) as proof of ((Xp1 Xx0) Xy0)
% Found (fun (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2))))) as proof of ((Xp1 Xx0) Xy0)
% Found (fun (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2))))) as proof of (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0))
% Found (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2))))) as proof of (forall (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))
% Found (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2))))) as proof of (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))
% Found ((conj00 (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2) as proof of ((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (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 (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2) as proof of ((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2) as proof of ((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2) as proof of ((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))
% Found (x3 ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2)) as proof of ((Xp1 Xx) Xy)
% Found ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2)) as proof of ((Xp1 Xx) Xy)
% Found (fun (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2))) as proof of ((Xp1 Xx) Xy)
% Found (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2))) as proof of ((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((Xp1 Xx) Xy))
% Found (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2))) as proof of ((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->((Xp1 Xx) Xy)))
% Found (and_rect00 (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2)))) as proof of ((Xp1 Xx) Xy)
% Found ((and_rect0 ((Xp1 Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2)))) as proof of ((Xp1 Xx) Xy)
% Found (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x1) x0)) ((Xp1 Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2)))) as proof of ((Xp1 Xx) Xy)
% Found (fun (x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x1) x0)) ((Xp1 Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2))))) as proof of ((Xp1 Xx) Xy)
% Found (fun (Xp1:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x1) x0)) ((Xp1 Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2))))) as proof of (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy))
% Found (fun (x:(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x1) x0)) ((Xp1 Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2))))) as proof of (forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x1) x0)) ((Xp1 Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2))))) as proof of ((forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x1) x0)) ((Xp1 Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2))))) as proof of (forall (Xy:a), ((forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x1) x0)) ((Xp1 Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2))))) as proof of (forall (Xx:a) (Xy:a), ((forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x1) x0)) ((Xp1 Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2))))) as proof of (forall (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x1) x0)) ((Xp1 Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2))))) as proof of (forall (R:(a->(a->Prop))) (S:(a->(a->Prop))) (Xx:a) (Xy:a), ((forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))->(forall (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (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 (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x1) x0)) ((Xp1 Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2)))))
% Time elapsed = 123.648183s
% node=11330 cost=1471.000000 depth=45
% ::::::::::::::::::::::
% % 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 (Xp1:(a->(a->Prop))), (((and (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))->((Xp1 Xx) Xy)))) (Xp1:(a->(a->Prop))) (x0:((and (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))))=> (((fun (P:Type) (x1:((forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))->((forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))->P)))=> (((((and_rect (forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) P) x1) x0)) ((Xp1 Xx) Xy)) (fun (x1:(forall (Xx0:a) (Xy0:a), (((or ((R Xx0) Xy0)) ((S Xx0) Xy0))->((Xp1 Xx0) Xy0)))) (x2:(forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz))))=> ((x Xp1) ((((conj (forall (Xx0:a) (Xy0:a), (((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))->((Xp1 Xx0) Xy0)))) (forall (Xx0:a) (Xy0:a) (Xz:a), (((and ((Xp1 Xx0) Xy0)) ((Xp1 Xy0) Xz))->((Xp1 Xx0) Xz)))) (fun (Xx0:a) (Xy0:a) (x4:((or (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))))=> ((((fun (P:Prop) (x5:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P)) (x6:((forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))->P))=> ((((((or_ind (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) (forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0)))) P) x5) x6) x4)) ((Xp1 Xx0) Xy0)) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((R Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((R Xx1) Xy1))=> (((x1 Xx1) Xy1) ((fun (B:Prop)=> (((or_introl ((R Xx1) Xy1)) B) x6)) ((S Xx1) Xy1))))) x2)))) (fun (x5:(forall (Xp10:(a->(a->Prop))), (((and (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp10 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp10 Xx1) Xy1)) ((Xp10 Xy1) Xz))->((Xp10 Xx1) Xz))))->((Xp10 Xx0) Xy0))))=> ((x5 Xp1) ((((conj (forall (Xx1:a) (Xy1:a), (((S Xx1) Xy1)->((Xp1 Xx1) Xy1)))) (forall (Xx1:a) (Xy1:a) (Xz:a), (((and ((Xp1 Xx1) Xy1)) ((Xp1 Xy1) Xz))->((Xp1 Xx1) Xz)))) (fun (Xx1:a) (Xy1:a) (x6:((S Xx1) Xy1))=> (((x1 Xx1) Xy1) (((or_intror ((R Xx1) Xy1)) ((S Xx1) Xy1)) x6)))) x2)))))) x2)))))
% % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p
% EOF
%------------------------------------------------------------------------------