%------------------------------------------------------------------------------ % 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 %------------------------------------------------------------------------------