%------------------------------------------------------------------------------
% File     : cocATP---0.2.0
% Problem  : SEU463^1 : TPTP v6.1.0. Released v3.6.0.
% Transfm  : none
% Format   : tptp:raw
% Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p

% Computer : n179.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 13:32:15 EDT 2014

% Result   : Timeout 300.05s
% Output   : None 
% Verified : 
% SZS Type : None (Parsing solution fails)
% Syntax   : Number of formulae    : 0

% Comments : 
%------------------------------------------------------------------------------
%----NO SOLUTION OUTPUT BY SYSTEM
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% % Problem  : SEU463^1 : TPTP v6.1.0. Released v3.6.0.
% % Command  : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p
% % Computer : n179.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 10:16:31 CDT 2014
% % CPUTime  : 300.05 
% Python 2.7.5
% Using paths ['/home/cristobal/cocATP/CASC/TPTP/', '/export/starexec/sandbox/benchmark/', '/export/starexec/sandbox/benchmark/']
% Failed to open /home/cristobal/cocATP/CASC/TPTP/Axioms/SET009^0.ax, trying next directory
% FOF formula (<kernel.Constant object at 0x236cfc8>, <kernel.DependentProduct object at 0x234d5f0>) of role type named subrel_type
% Using role type
% Declaring subrel:((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->Prop))
% FOF formula (((eq ((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->Prop))) subrel) (fun (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((S X) Y))))) of role definition named subrel
% A new definition: (((eq ((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->Prop))) subrel) (fun (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((S X) Y)))))
% Defined: subrel:=(fun (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((S X) Y))))
% FOF formula (<kernel.Constant object at 0x236cfc8>, <kernel.DependentProduct object at 0x236c200>) of role type named inv_type
% Using role type
% Declaring inv:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) inv) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((R Y) X))) of role definition named inverse
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) inv) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((R Y) X)))
% Defined: inv:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((R Y) X))
% FOF formula (<kernel.Constant object at 0x236cfc8>, <kernel.DependentProduct object at 0x218a830>) of role type named idem_type
% Using role type
% Declaring idem:(((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)
% FOF formula (((eq (((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)) idem) (fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), (((eq (fofType->(fofType->Prop))) (F (F R))) (F R))))) of role definition named idempotent
% A new definition: (((eq (((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)) idem) (fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), (((eq (fofType->(fofType->Prop))) (F (F R))) (F R)))))
% Defined: idem:=(fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), (((eq (fofType->(fofType->Prop))) (F (F R))) (F R))))
% FOF formula (<kernel.Constant object at 0x236c200>, <kernel.DependentProduct object at 0x218a440>) of role type named infl_type
% Using role type
% Declaring infl:(((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)
% FOF formula (((eq (((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)) infl) (fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), ((subrel R) (F R))))) of role definition named inflationary
% A new definition: (((eq (((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)) infl) (fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), ((subrel R) (F R)))))
% Defined: infl:=(fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), ((subrel R) (F R))))
% FOF formula (<kernel.Constant object at 0x218a440>, <kernel.DependentProduct object at 0x218a7a0>) of role type named mono_type
% Using role type
% Declaring mono:(((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)
% FOF formula (((eq (((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)) mono) (fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop))), (((subrel R) S)->((subrel (F R)) (F S)))))) of role definition named monotonic
% A new definition: (((eq (((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop)) mono) (fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop))), (((subrel R) S)->((subrel (F R)) (F S))))))
% Defined: mono:=(fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop))), (((subrel R) S)->((subrel (F R)) (F S)))))
% FOF formula (<kernel.Constant object at 0x218a440>, <kernel.DependentProduct object at 0x218a710>) of role type named refl_type
% Using role type
% Declaring refl:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) refl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((R X) X)))) of role definition named reflexive
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) refl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((R X) X))))
% Defined: refl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((R X) X)))
% FOF formula (<kernel.Constant object at 0x218a5f0>, <kernel.DependentProduct object at 0x218ac20>) of role type named irrefl_type
% Using role type
% Declaring irrefl:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) irrefl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), (((R X) X)->False)))) of role definition named irreflexive
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) irrefl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), (((R X) X)->False))))
% Defined: irrefl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), (((R X) X)->False)))
% FOF formula (<kernel.Constant object at 0x2054d40>, <kernel.DependentProduct object at 0x218add0>) of role type named rc_type
% Using role type
% Declaring rc:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) rc) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or (((eq fofType) X) Y)) ((R X) Y)))) of role definition named reflexive_closure
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) rc) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or (((eq fofType) X) Y)) ((R X) Y))))
% Defined: rc:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or (((eq fofType) X) Y)) ((R X) Y)))
% FOF formula (<kernel.Constant object at 0x218a3b0>, <kernel.DependentProduct object at 0x218a830>) of role type named symm_type
% Using role type
% Declaring symm:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) symm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((R Y) X))))) of role definition named symmetric
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) symm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((R Y) X)))))
% Defined: symm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((R Y) X))))
% FOF formula (<kernel.Constant object at 0x218aef0>, <kernel.DependentProduct object at 0x218a488>) of role type named antisymm_type
% Using role type
% Declaring antisymm:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) antisymm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((and ((R X) Y)) ((R Y) X))->(((eq fofType) X) Y))))) of role definition named antisymmetric
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) antisymm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((and ((R X) Y)) ((R Y) X))->(((eq fofType) X) Y)))))
% Defined: antisymm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((and ((R X) Y)) ((R Y) X))->(((eq fofType) X) Y))))
% FOF formula (<kernel.Constant object at 0x218a3b0>, <kernel.DependentProduct object at 0x2180680>) of role type named asymm_type
% Using role type
% Declaring asymm:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) asymm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->(((R Y) X)->False))))) of role definition named asymmetric
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) asymm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->(((R Y) X)->False)))))
% Defined: asymm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->(((R Y) X)->False))))
% FOF formula (<kernel.Constant object at 0x218aef0>, <kernel.DependentProduct object at 0x2180248>) of role type named sc_type
% Using role type
% Declaring sc:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) sc) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or ((R Y) X)) ((R X) Y)))) of role definition named symmetric_closure
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) sc) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or ((R Y) X)) ((R X) Y))))
% Defined: sc:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or ((R Y) X)) ((R X) Y)))
% FOF formula (<kernel.Constant object at 0x21807e8>, <kernel.DependentProduct object at 0x2180488>) of role type named trans_type
% Using role type
% Declaring trans:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) trans) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Y)) ((R Y) Z))->((R X) Z))))) of role definition named transitive
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) trans) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Y)) ((R Y) Z))->((R X) Z)))))
% Defined: trans:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Y)) ((R Y) Z))->((R X) Z))))
% FOF formula (<kernel.Constant object at 0x2180518>, <kernel.DependentProduct object at 0x2180908>) of role type named tc_type
% Using role type
% Declaring tc:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) tc) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> (forall (S:(fofType->(fofType->Prop))), (((and (trans S)) ((subrel R) S))->((S X) Y))))) of role definition named transitive_closure
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) tc) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> (forall (S:(fofType->(fofType->Prop))), (((and (trans S)) ((subrel R) S))->((S X) Y)))))
% Defined: tc:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> (forall (S:(fofType->(fofType->Prop))), (((and (trans S)) ((subrel R) S))->((S X) Y))))
% FOF formula (<kernel.Constant object at 0x2180710>, <kernel.DependentProduct object at 0x2180440>) of role type named trc_type
% Using role type
% Declaring trc:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) trc) (fun (R:(fofType->(fofType->Prop)))=> (rc (tc R)))) of role definition named transitive_reflexive_closure
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) trc) (fun (R:(fofType->(fofType->Prop)))=> (rc (tc R))))
% Defined: trc:=(fun (R:(fofType->(fofType->Prop)))=> (rc (tc R)))
% FOF formula (<kernel.Constant object at 0x21800e0>, <kernel.DependentProduct object at 0x21809e0>) of role type named trsc_type
% Using role type
% Declaring trsc:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) trsc) (fun (R:(fofType->(fofType->Prop)))=> (sc (rc (tc R))))) of role definition named transitive_reflexive_symmetric_closure
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) trsc) (fun (R:(fofType->(fofType->Prop)))=> (sc (rc (tc R)))))
% Defined: trsc:=(fun (R:(fofType->(fofType->Prop)))=> (sc (rc (tc R))))
% FOF formula (<kernel.Constant object at 0x21800e0>, <kernel.DependentProduct object at 0x21805f0>) of role type named po_type
% Using role type
% Declaring po:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) po) (fun (R:(fofType->(fofType->Prop)))=> ((and ((and (refl R)) (antisymm R))) (trans R)))) of role definition named partial_order
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) po) (fun (R:(fofType->(fofType->Prop)))=> ((and ((and (refl R)) (antisymm R))) (trans R))))
% Defined: po:=(fun (R:(fofType->(fofType->Prop)))=> ((and ((and (refl R)) (antisymm R))) (trans R)))
% FOF formula (<kernel.Constant object at 0x2180200>, <kernel.DependentProduct object at 0x21805a8>) of role type named so_type
% Using role type
% Declaring so:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) so) (fun (R:(fofType->(fofType->Prop)))=> ((and (asymm R)) (trans R)))) of role definition named strict_order
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) so) (fun (R:(fofType->(fofType->Prop)))=> ((and (asymm R)) (trans R))))
% Defined: so:=(fun (R:(fofType->(fofType->Prop)))=> ((and (asymm R)) (trans R)))
% FOF formula (<kernel.Constant object at 0x21805f0>, <kernel.DependentProduct object at 0x21804d0>) of role type named total_type
% Using role type
% Declaring total:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) total) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((or ((or (((eq fofType) X) Y)) ((R X) Y))) ((R Y) X))))) of role definition named total
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) total) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((or ((or (((eq fofType) X) Y)) ((R X) Y))) ((R Y) X)))))
% Defined: total:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((or ((or (((eq fofType) X) Y)) ((R X) Y))) ((R Y) X))))
% FOF formula (<kernel.Constant object at 0x21804d0>, <kernel.DependentProduct object at 0x2180b48>) of role type named term_type
% Using role type
% Declaring term:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) term) (fun (R:(fofType->(fofType->Prop)))=> (forall (A:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> (A X)))->((ex fofType) (fun (X:fofType)=> ((and (A X)) (forall (Y:fofType), ((A Y)->(((R X) Y)->False)))))))))) of role definition named terminating
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) term) (fun (R:(fofType->(fofType->Prop)))=> (forall (A:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> (A X)))->((ex fofType) (fun (X:fofType)=> ((and (A X)) (forall (Y:fofType), ((A Y)->(((R X) Y)->False))))))))))
% Defined: term:=(fun (R:(fofType->(fofType->Prop)))=> (forall (A:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> (A X)))->((ex fofType) (fun (X:fofType)=> ((and (A X)) (forall (Y:fofType), ((A Y)->(((R X) Y)->False)))))))))
% FOF formula (<kernel.Constant object at 0x2180bd8>, <kernel.DependentProduct object at 0x2180cf8>) of role type named ind_type
% Using role type
% Declaring ind:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) ind) (fun (R:(fofType->(fofType->Prop)))=> (forall (P:(fofType->Prop)), ((forall (X:fofType), ((forall (Y:fofType), ((((tc R) X) Y)->(P Y)))->(P X)))->(forall (X:fofType), (P X)))))) of role definition named satisfying_the_induction_principle
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) ind) (fun (R:(fofType->(fofType->Prop)))=> (forall (P:(fofType->Prop)), ((forall (X:fofType), ((forall (Y:fofType), ((((tc R) X) Y)->(P Y)))->(P X)))->(forall (X:fofType), (P X))))))
% Defined: ind:=(fun (R:(fofType->(fofType->Prop)))=> (forall (P:(fofType->Prop)), ((forall (X:fofType), ((forall (Y:fofType), ((((tc R) X) Y)->(P Y)))->(P X)))->(forall (X:fofType), (P X)))))
% FOF formula (<kernel.Constant object at 0x21801b8>, <kernel.DependentProduct object at 0x2180bd8>) of role type named innf_type
% Using role type
% Declaring innf:((fofType->(fofType->Prop))->(fofType->Prop))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->Prop))) innf) (fun (R:(fofType->(fofType->Prop))) (X:fofType)=> (((ex fofType) (fun (Y:fofType)=> ((R X) Y)))->False))) of role definition named in_normal_form
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->Prop))) innf) (fun (R:(fofType->(fofType->Prop))) (X:fofType)=> (((ex fofType) (fun (Y:fofType)=> ((R X) Y)))->False)))
% Defined: innf:=(fun (R:(fofType->(fofType->Prop))) (X:fofType)=> (((ex fofType) (fun (Y:fofType)=> ((R X) Y)))->False))
% FOF formula (<kernel.Constant object at 0x2180cf8>, <kernel.DependentProduct object at 0x21800e0>) of role type named nfof_type
% Using role type
% Declaring nfof:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) nfof) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((and (((trc R) Y) X)) ((innf R) X)))) of role definition named normal_form_of
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) nfof) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((and (((trc R) Y) X)) ((innf R) X))))
% Defined: nfof:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((and (((trc R) Y) X)) ((innf R) X)))
% FOF formula (<kernel.Constant object at 0x21801b8>, <kernel.DependentProduct object at 0x2180d88>) of role type named norm_type
% Using role type
% Declaring norm:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) norm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((ex fofType) (fun (Y:fofType)=> (((nfof R) Y) X)))))) of role definition named normalizing
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) norm) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((ex fofType) (fun (Y:fofType)=> (((nfof R) Y) X))))))
% Defined: norm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((ex fofType) (fun (Y:fofType)=> (((nfof R) Y) X)))))
% FOF formula (<kernel.Constant object at 0x21801b8>, <kernel.DependentProduct object at 0x2180f80>) of role type named join_type
% Using role type
% Declaring join:((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))
% FOF formula (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) join) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((ex fofType) (fun (Z:fofType)=> ((and (((trc R) X) Z)) (((trc R) Y) Z)))))) of role definition named joinable
% A new definition: (((eq ((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))) join) (fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((ex fofType) (fun (Z:fofType)=> ((and (((trc R) X) Z)) (((trc R) Y) Z))))))
% Defined: join:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((ex fofType) (fun (Z:fofType)=> ((and (((trc R) X) Z)) (((trc R) Y) Z)))))
% FOF formula (<kernel.Constant object at 0x2180680>, <kernel.DependentProduct object at 0x235a3f8>) of role type named lconfl_type
% Using role type
% Declaring lconfl:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) lconfl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) ((R X) Y))->(((join R) Z) Y))))) of role definition named locally_confluent
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) lconfl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) ((R X) Y))->(((join R) Z) Y)))))
% Defined: lconfl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) ((R X) Y))->(((join R) Z) Y))))
% FOF formula (<kernel.Constant object at 0x21801b8>, <kernel.DependentProduct object at 0x235a050>) of role type named sconfl_type
% Using role type
% Declaring sconfl:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) sconfl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) (((trc R) X) Y))->(((join R) Z) Y))))) of role definition named semi_confluent
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) sconfl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) (((trc R) X) Y))->(((join R) Z) Y)))))
% Defined: sconfl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) (((trc R) X) Y))->(((join R) Z) Y))))
% FOF formula (<kernel.Constant object at 0x235a3f8>, <kernel.DependentProduct object at 0x235a440>) of role type named confl_type
% Using role type
% Declaring confl:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) confl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and (((trc R) X) Z)) (((trc R) X) Y))->(((join R) Z) Y))))) of role definition named confluent
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) confl) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and (((trc R) X) Z)) (((trc R) X) Y))->(((join R) Z) Y)))))
% Defined: confl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and (((trc R) X) Z)) (((trc R) X) Y))->(((join R) Z) Y))))
% FOF formula (<kernel.Constant object at 0x235a050>, <kernel.DependentProduct object at 0x235a320>) of role type named cr_type
% Using role type
% Declaring cr:((fofType->(fofType->Prop))->Prop)
% FOF formula (((eq ((fofType->(fofType->Prop))->Prop)) cr) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((((trsc R) X) Y)->(((join R) X) Y))))) of role definition named church_rosser
% A new definition: (((eq ((fofType->(fofType->Prop))->Prop)) cr) (fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((((trsc R) X) Y)->(((join R) X) Y)))))
% Defined: cr:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((((trsc R) X) Y)->(((join R) X) Y))))
% FOF formula (forall (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType) (Z:fofType) (S:(fofType->(fofType->Prop))), (((and ((and ((and (trans S)) ((subrel R) S))) (((tc R) X) Y))) (((tc R) Y) Z))->((S X) Z))) of role conjecture named transitive_closure_is_transitive3
% Conjecture to prove = (forall (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType) (Z:fofType) (S:(fofType->(fofType->Prop))), (((and ((and ((and (trans S)) ((subrel R) S))) (((tc R) X) Y))) (((tc R) Y) Z))->((S X) Z))):Prop
% Parameter fofType_DUMMY:fofType.
% We need to prove ['(forall (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType) (Z:fofType) (S:(fofType->(fofType->Prop))), (((and ((and ((and (trans S)) ((subrel R) S))) (((tc R) X) Y))) (((tc R) Y) Z))->((S X) Z)))']
% Parameter fofType:Type.
% Definition subrel:=(fun (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((S X) Y)))):((fofType->(fofType->Prop))->((fofType->(fofType->Prop))->Prop)).
% Definition inv:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((R Y) X)):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition idem:=(fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), (((eq (fofType->(fofType->Prop))) (F (F R))) (F R)))):(((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop).
% Definition infl:=(fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))), ((subrel R) (F R)))):(((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop).
% Definition mono:=(fun (F:((fofType->(fofType->Prop))->(fofType->(fofType->Prop))))=> (forall (R:(fofType->(fofType->Prop))) (S:(fofType->(fofType->Prop))), (((subrel R) S)->((subrel (F R)) (F S))))):(((fofType->(fofType->Prop))->(fofType->(fofType->Prop)))->Prop).
% Definition refl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((R X) X))):((fofType->(fofType->Prop))->Prop).
% Definition irrefl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), (((R X) X)->False))):((fofType->(fofType->Prop))->Prop).
% Definition rc:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or (((eq fofType) X) Y)) ((R X) Y))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition symm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->((R Y) X)))):((fofType->(fofType->Prop))->Prop).
% Definition antisymm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((and ((R X) Y)) ((R Y) X))->(((eq fofType) X) Y)))):((fofType->(fofType->Prop))->Prop).
% Definition asymm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), (((R X) Y)->(((R Y) X)->False)))):((fofType->(fofType->Prop))->Prop).
% Definition sc:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((or ((R Y) X)) ((R X) Y))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition trans:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Y)) ((R Y) Z))->((R X) Z)))):((fofType->(fofType->Prop))->Prop).
% Definition tc:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> (forall (S:(fofType->(fofType->Prop))), (((and (trans S)) ((subrel R) S))->((S X) Y)))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition trc:=(fun (R:(fofType->(fofType->Prop)))=> (rc (tc R))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition trsc:=(fun (R:(fofType->(fofType->Prop)))=> (sc (rc (tc R)))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition po:=(fun (R:(fofType->(fofType->Prop)))=> ((and ((and (refl R)) (antisymm R))) (trans R))):((fofType->(fofType->Prop))->Prop).
% Definition so:=(fun (R:(fofType->(fofType->Prop)))=> ((and (asymm R)) (trans R))):((fofType->(fofType->Prop))->Prop).
% Definition total:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((or ((or (((eq fofType) X) Y)) ((R X) Y))) ((R Y) X)))):((fofType->(fofType->Prop))->Prop).
% Definition term:=(fun (R:(fofType->(fofType->Prop)))=> (forall (A:(fofType->Prop)), (((ex fofType) (fun (X:fofType)=> (A X)))->((ex fofType) (fun (X:fofType)=> ((and (A X)) (forall (Y:fofType), ((A Y)->(((R X) Y)->False))))))))):((fofType->(fofType->Prop))->Prop).
% Definition ind:=(fun (R:(fofType->(fofType->Prop)))=> (forall (P:(fofType->Prop)), ((forall (X:fofType), ((forall (Y:fofType), ((((tc R) X) Y)->(P Y)))->(P X)))->(forall (X:fofType), (P X))))):((fofType->(fofType->Prop))->Prop).
% Definition innf:=(fun (R:(fofType->(fofType->Prop))) (X:fofType)=> (((ex fofType) (fun (Y:fofType)=> ((R X) Y)))->False)):((fofType->(fofType->Prop))->(fofType->Prop)).
% Definition nfof:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((and (((trc R) Y) X)) ((innf R) X))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition norm:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType), ((ex fofType) (fun (Y:fofType)=> (((nfof R) Y) X))))):((fofType->(fofType->Prop))->Prop).
% Definition join:=(fun (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType)=> ((ex fofType) (fun (Z:fofType)=> ((and (((trc R) X) Z)) (((trc R) Y) Z))))):((fofType->(fofType->Prop))->(fofType->(fofType->Prop))).
% Definition lconfl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) ((R X) Y))->(((join R) Z) Y)))):((fofType->(fofType->Prop))->Prop).
% Definition sconfl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and ((R X) Z)) (((trc R) X) Y))->(((join R) Z) Y)))):((fofType->(fofType->Prop))->Prop).
% Definition confl:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType) (Z:fofType), (((and (((trc R) X) Z)) (((trc R) X) Y))->(((join R) Z) Y)))):((fofType->(fofType->Prop))->Prop).
% Definition cr:=(fun (R:(fofType->(fofType->Prop)))=> (forall (X:fofType) (Y:fofType), ((((trsc R) X) Y)->(((join R) X) Y)))):((fofType->(fofType->Prop))->Prop).
% Trying to prove (forall (R:(fofType->(fofType->Prop))) (X:fofType) (Y:fofType) (Z:fofType) (S:(fofType->(fofType->Prop))), (((and ((and ((and (trans S)) ((subrel R) S))) (((tc R) X) Y))) (((tc R) Y) Z))->((S X) Z)))
% Found x4:((S X) Z0)
% Found (fun (x4:((S X) Z0))=> x4) as proof of ((S X) Z0)
% Found (fun (x3:((S X) Y0)) (x4:((S X) Z0))=> x4) as proof of (((S X) Z0)->((S X) Z0))
% Found (fun (x3:((S X) Y0)) (x4:((S X) Z0))=> x4) as proof of (((S X) Y0)->(((S X) Z0)->((S X) Z0)))
% Found (and_rect10 (fun (x3:((S X) Y0)) (x4:((S X) Z0))=> x4)) as proof of (((fun (x4:fofType)=> (S X)) X0) Z0)
% Found ((and_rect1 ((S X) Z0)) (fun (x3:((S X) Y0)) (x4:((S X) Z0))=> x4)) as proof of (((fun (x4:fofType)=> (S X)) X0) Z0)
% Found (((fun (P:Type) (x3:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x3) x2)) ((S X) Z0)) (fun (x3:((S X) Y0)) (x4:((S X) Z0))=> x4)) as proof of (((fun (x4:fofType)=> (S X)) X0) Z0)
% Found (fun (x2:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x3:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x3) x2)) ((S X) Z0)) (fun (x3:((S X) Y0)) (x4:((S X) Z0))=> x4))) as proof of (((fun (x4:fofType)=> (S X)) X0) Z0)
% Found (fun (Z0:fofType) (x2:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x3:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x3) x2)) ((S X) Z0)) (fun (x3:((S X) Y0)) (x4:((S X) Z0))=> x4))) as proof of (((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0))->(((fun (x4:fofType)=> (S X)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x2:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x3:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x3) x2)) ((S X) Z0)) (fun (x3:((S X) Y0)) (x4:((S X) Z0))=> x4))) as proof of (forall (Z:fofType), (((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z))->(((fun (x4:fofType)=> (S X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x2:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x3:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x3) x2)) ((S X) Z0)) (fun (x3:((S X) Y0)) (x4:((S X) Z0))=> x4))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x4:fofType)=> (S X)) X0) Y)) (((fun (x4:fofType)=> (S X)) Y) Z))->(((fun (x4:fofType)=> (S X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x2:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x3:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x3) x2)) ((S X) Z0)) (fun (x3:((S X) Y0)) (x4:((S X) Z0))=> x4))) as proof of (trans (fun (x4:fofType)=> (S X)))
% Found x4:(((inv S) Z0) X)
% Found (fun (x4:(((inv S) Z0) X))=> x4) as proof of (((inv S) Z0) X)
% Found (fun (x3:(((inv S) Y0) X)) (x4:(((inv S) Z0) X))=> x4) as proof of ((((inv S) Z0) X)->(((inv S) Z0) X))
% Found (fun (x3:(((inv S) Y0) X)) (x4:(((inv S) Z0) X))=> x4) as proof of ((((inv S) Y0) X)->((((inv S) Z0) X)->(((inv S) Z0) X)))
% Found (and_rect10 (fun (x3:(((inv S) Y0) X)) (x4:(((inv S) Z0) X))=> x4)) as proof of (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0)
% Found ((and_rect1 (((inv S) Z0) X)) (fun (x3:(((inv S) Y0) X)) (x4:(((inv S) Z0) X))=> x4)) as proof of (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0)
% Found (((fun (P:Type) (x3:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x3) x2)) (((inv S) Z0) X)) (fun (x3:(((inv S) Y0) X)) (x4:(((inv S) Z0) X))=> x4)) as proof of (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0)
% Found (fun (x2:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x3:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x3) x2)) (((inv S) Z0) X)) (fun (x3:(((inv S) Y0) X)) (x4:(((inv S) Z0) X))=> x4))) as proof of (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0)
% Found (fun (Z0:fofType) (x2:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x3:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x3) x2)) (((inv S) Z0) X)) (fun (x3:(((inv S) Y0) X)) (x4:(((inv S) Z0) X))=> x4))) as proof of (((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0))->(((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x2:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x3:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x3) x2)) (((inv S) Z0) X)) (fun (x3:(((inv S) Y0) X)) (x4:(((inv S) Z0) X))=> x4))) as proof of (forall (Z:fofType), (((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z))->(((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x2:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x3:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x3) x2)) (((inv S) Z0) X)) (fun (x3:(((inv S) Y0) X)) (x4:(((inv S) Z0) X))=> x4))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y) Z))->(((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x2:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x3:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x3) x2)) (((inv S) Z0) X)) (fun (x3:(((inv S) Y0) X)) (x4:(((inv S) Z0) X))=> x4))) as proof of (trans (fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)))
% Found x5:((S X0) Z)
% Found (fun (x6:((S Y0) Z))=> x5) as proof of ((S X0) Z)
% Found (fun (x5:((S X0) Z)) (x6:((S Y0) Z))=> x5) as proof of (((S Y0) Z)->((S X0) Z))
% Found (fun (x5:((S X0) Z)) (x6:((S Y0) Z))=> x5) as proof of (((S X0) Z)->(((S Y0) Z)->((S X0) Z)))
% Found (and_rect20 (fun (x5:((S X0) Z)) (x6:((S Y0) Z))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Z0)
% Found ((and_rect2 ((S X0) Z)) (fun (x5:((S X0) Z)) (x6:((S Y0) Z))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Z0)
% Found (((fun (P:Type) (x5:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x5) x4)) ((S X0) Z)) (fun (x5:((S X0) Z)) (x6:((S Y0) Z))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x5) x4)) ((S X0) Z)) (fun (x5:((S X0) Z)) (x6:((S Y0) Z))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x5) x4)) ((S X0) Z)) (fun (x5:((S X0) Z)) (x6:((S Y0) Z))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x5) x4)) ((S X0) Z)) (fun (x5:((S X0) Z)) (x6:((S Y0) Z))=> x5))) as proof of (forall (Z0:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x5) x4)) ((S X0) Z)) (fun (x5:((S X0) Z)) (x6:((S Y0) Z))=> x5))) as proof of (forall (Y:fofType) (Z0:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x5) x4)) ((S X0) Z)) (fun (x5:((S X0) Z)) (x6:((S Y0) Z))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((S x6) Z)))
% Found x6:((S X) Z0)
% Found (fun (x6:((S X) Z0))=> x6) as proof of ((S X) Z0)
% Found (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6) as proof of (((S X) Z0)->((S X) Z0))
% Found (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6) as proof of (((S X) Y0)->(((S X) Z0)->((S X) Z0)))
% Found (and_rect20 (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6)) as proof of (((fun (x6:fofType)=> (S X)) X0) Z0)
% Found ((and_rect2 ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6)) as proof of (((fun (x6:fofType)=> (S X)) X0) Z0)
% Found (((fun (P:Type) (x5:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x5) x4)) ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6)) as proof of (((fun (x6:fofType)=> (S X)) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x5) x4)) ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6))) as proof of (((fun (x6:fofType)=> (S X)) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x5) x4)) ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6))) as proof of (((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z0))->(((fun (x6:fofType)=> (S X)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x5) x4)) ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z))->(((fun (x6:fofType)=> (S X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x5) x4)) ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType)=> (S X)) X0) Y)) (((fun (x6:fofType)=> (S X)) Y) Z))->(((fun (x6:fofType)=> (S X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x5) x4)) ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6))) as proof of (trans (fun (x6:fofType)=> (S X)))
% Found x6:((S X) Z0)
% Found (fun (x6:((S X) Z0))=> x6) as proof of ((S X) Z0)
% Found (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6) as proof of (((S X) Z0)->((S X) Z0))
% Found (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6) as proof of (((S X) Y0)->(((S X) Z0)->((S X) Z0)))
% Found (and_rect20 (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6)) as proof of (((fun (x40:fofType)=> (S X)) X0) Z0)
% Found ((and_rect2 ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6)) as proof of (((fun (x40:fofType)=> (S X)) X0) Z0)
% Found (((fun (P:Type) (x5:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x5) x4)) ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6)) as proof of (((fun (x40:fofType)=> (S X)) X0) Z0)
% Found (fun (x4:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x5) x4)) ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6))) as proof of (((fun (x40:fofType)=> (S X)) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x5) x4)) ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6))) as proof of (((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0))->(((fun (x4:fofType)=> (S X)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x5) x4)) ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z))->(((fun (x4:fofType)=> (S X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x5) x4)) ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x4:fofType)=> (S X)) X0) Y)) (((fun (x4:fofType)=> (S X)) Y) Z))->(((fun (x4:fofType)=> (S X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x5:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x5) x4)) ((S X) Z0)) (fun (x5:((S X) Y0)) (x6:((S X) Z0))=> x6))) as proof of (trans (fun (x4:fofType)=> (S X)))
% Found x6:(((inv S) Z0) X)
% Found (fun (x6:(((inv S) Z0) X))=> x6) as proof of (((inv S) Z0) X)
% Found (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6) as proof of ((((inv S) Z0) X)->(((inv S) Z0) X))
% Found (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6) as proof of ((((inv S) Y0) X)->((((inv S) Z0) X)->(((inv S) Z0) X)))
% Found (and_rect20 (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z0)
% Found ((and_rect2 (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z0)
% Found (((fun (P:Type) (x5:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x5) x4)) (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x5) x4)) (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6))) as proof of (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x5) x4)) (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x5) x4)) (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x5) x4)) (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x5) x4)) (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)))
% Found x5:(((inv S) Z) X0)
% Found (fun (x6:(((inv S) Z) Y0))=> x5) as proof of (((inv S) Z) X0)
% Found (fun (x5:(((inv S) Z) X0)) (x6:(((inv S) Z) Y0))=> x5) as proof of ((((inv S) Z) Y0)->(((inv S) Z) X0))
% Found (fun (x5:(((inv S) Z) X0)) (x6:(((inv S) Z) Y0))=> x5) as proof of ((((inv S) Z) X0)->((((inv S) Z) Y0)->(((inv S) Z) X0)))
% Found (and_rect20 (fun (x5:(((inv S) Z) X0)) (x6:(((inv S) Z) Y0))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Z0)
% Found ((and_rect2 (((inv S) Z) X0)) (fun (x5:(((inv S) Z) X0)) (x6:(((inv S) Z) Y0))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Z0)
% Found (((fun (P:Type) (x5:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x5) x4)) (((inv S) Z) X0)) (fun (x5:(((inv S) Z) X0)) (x6:(((inv S) Z) Y0))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x5) x4)) (((inv S) Z) X0)) (fun (x5:(((inv S) Z) X0)) (x6:(((inv S) Z) Y0))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x5) x4)) (((inv S) Z) X0)) (fun (x5:(((inv S) Z) X0)) (x6:(((inv S) Z) Y0))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x5) x4)) (((inv S) Z) X0)) (fun (x5:(((inv S) Z) X0)) (x6:(((inv S) Z) Y0))=> x5))) as proof of (forall (Z0:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x5) x4)) (((inv S) Z) X0)) (fun (x5:(((inv S) Z) X0)) (x6:(((inv S) Z) Y0))=> x5))) as proof of (forall (Y:fofType) (Z0:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y) Z0))->(((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x5) x4)) (((inv S) Z) X0)) (fun (x5:(((inv S) Z) X0)) (x6:(((inv S) Z) Y0))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)))
% Found x6:(((inv S) Z0) X)
% Found (fun (x6:(((inv S) Z0) X))=> x6) as proof of (((inv S) Z0) X)
% Found (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6) as proof of ((((inv S) Z0) X)->(((inv S) Z0) X))
% Found (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6) as proof of ((((inv S) Y0) X)->((((inv S) Z0) X)->(((inv S) Z0) X)))
% Found (and_rect20 (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6)) as proof of (((fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0)
% Found ((and_rect2 (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6)) as proof of (((fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0)
% Found (((fun (P:Type) (x5:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x5) x4)) (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6)) as proof of (((fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0)
% Found (fun (x4:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x5) x4)) (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6))) as proof of (((fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x5) x4)) (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6))) as proof of (((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0))->(((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x5) x4)) (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z))->(((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x5) x4)) (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y) Z))->(((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x5:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x5) x4)) (((inv S) Z0) X)) (fun (x5:(((inv S) Y0) X)) (x6:(((inv S) Z0) X))=> x6))) as proof of (trans (fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)))
% Found x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))
% Found (fun (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))))=> x6) as proof of ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))))=> x6) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))->((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))))=> x6) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))->((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))))
% Found (and_rect20 (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Z0)
% Found ((and_rect2 ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))))=> x6))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))))=> x6))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))))=> x6))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50))))))
% Found x5:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))
% Found (fun (x6:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x5) as proof of ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))
% Found (fun (x5:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x6:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x5) as proof of (((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0)))))
% Found (fun (x5:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x6:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x5) as proof of (((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))
% Found (and_rect20 (fun (x5:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x6:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Z0)
% Found ((and_rect2 ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x6:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) ((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))) P) x5) x4)) ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x6:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) ((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))) P) x5) x4)) ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x6:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) ((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))) P) x5) x4)) ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x6:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) ((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))) P) x5) x4)) ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x6:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) ((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))) P) x5) x4)) ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x6:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) ((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))) P) x5) x4)) ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x6:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))
% Found x5:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))
% Found (fun (x6:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x5) as proof of ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))
% Found (fun (x5:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x6:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x5) as proof of (((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))
% Found (fun (x5:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x6:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x5) as proof of (((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))))
% Found (and_rect20 (fun (x5:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x6:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Z0)
% Found ((and_rect2 ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x6:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) ((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))) P) x5) x4)) ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x6:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) ((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))) P) x5) x4)) ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x6:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) ((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))) P) x5) x4)) ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x6:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) ((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))) P) x5) x4)) ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x6:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) ((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))) P) x5) x4)) ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x6:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) ((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))) P) x5) x4)) ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x5:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x6:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType)=> (S x6)))) ((subrel R) (fun (x40:fofType)=> (S x6))))))
% Found x6:((trans S)->(((subrel R) S)->((S X) Z0)))
% Found (fun (x6:((trans S)->(((subrel R) S)->((S X) Z0))))=> x6) as proof of ((trans S)->(((subrel R) S)->((S X) Z0)))
% Found (fun (x5:((trans S)->(((subrel R) S)->((S X) Y0)))) (x6:((trans S)->(((subrel R) S)->((S X) Z0))))=> x6) as proof of (((trans S)->(((subrel R) S)->((S X) Z0)))->((trans S)->(((subrel R) S)->((S X) Z0))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((S X) Y0)))) (x6:((trans S)->(((subrel R) S)->((S X) Z0))))=> x6) as proof of (((trans S)->(((subrel R) S)->((S X) Y0)))->(((trans S)->(((subrel R) S)->((S X) Z0)))->((trans S)->(((subrel R) S)->((S X) Z0)))))
% Found (and_rect30 (fun (x5:((trans S)->(((subrel R) S)->((S X) Y0)))) (x6:((trans S)->(((subrel R) S)->((S X) Z0))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Z0)
% Found ((and_rect3 ((trans S)->(((subrel R) S)->((S X) Z0)))) (fun (x5:((trans S)->(((subrel R) S)->((S X) Y0)))) (x6:((trans S)->(((subrel R) S)->((S X) Z0))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Z0)
% Found (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((S X) Y0)))->(((trans S)->(((subrel R) S)->((S X) Z0)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((S X) Y0)))) ((trans S)->(((subrel R) S)->((S X) Z0)))) P) x5) x4)) ((trans S)->(((subrel R) S)->((S X) Z0)))) (fun (x5:((trans S)->(((subrel R) S)->((S X) Y0)))) (x6:((trans S)->(((subrel R) S)->((S X) Z0))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((S X) Y0)))->(((trans S)->(((subrel R) S)->((S X) Z0)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((S X) Y0)))) ((trans S)->(((subrel R) S)->((S X) Z0)))) P) x5) x4)) ((trans S)->(((subrel R) S)->((S X) Z0)))) (fun (x5:((trans S)->(((subrel R) S)->((S X) Y0)))) (x6:((trans S)->(((subrel R) S)->((S X) Z0))))=> x6))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((S X) Y0)))->(((trans S)->(((subrel R) S)->((S X) Z0)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((S X) Y0)))) ((trans S)->(((subrel R) S)->((S X) Z0)))) P) x5) x4)) ((trans S)->(((subrel R) S)->((S X) Z0)))) (fun (x5:((trans S)->(((subrel R) S)->((S X) Y0)))) (x6:((trans S)->(((subrel R) S)->((S X) Z0))))=> x6))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((S X) Y0)))->(((trans S)->(((subrel R) S)->((S X) Z0)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((S X) Y0)))) ((trans S)->(((subrel R) S)->((S X) Z0)))) P) x5) x4)) ((trans S)->(((subrel R) S)->((S X) Z0)))) (fun (x5:((trans S)->(((subrel R) S)->((S X) Y0)))) (x6:((trans S)->(((subrel R) S)->((S X) Z0))))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((S X) Y0)))->(((trans S)->(((subrel R) S)->((S X) Z0)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((S X) Y0)))) ((trans S)->(((subrel R) S)->((S X) Z0)))) P) x5) x4)) ((trans S)->(((subrel R) S)->((S X) Z0)))) (fun (x5:((trans S)->(((subrel R) S)->((S X) Y0)))) (x6:((trans S)->(((subrel R) S)->((S X) Z0))))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((S X) Y0)))->(((trans S)->(((subrel R) S)->((S X) Z0)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((S X) Y0)))) ((trans S)->(((subrel R) S)->((S X) Z0)))) P) x5) x4)) ((trans S)->(((subrel R) S)->((S X) Z0)))) (fun (x5:((trans S)->(((subrel R) S)->((S X) Y0)))) (x6:((trans S)->(((subrel R) S)->((S X) Z0))))=> x6))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S X) x50)))))
% Found x5:((trans S)->(((subrel R) S)->((S X0) Z)))
% Found (fun (x6:((trans S)->(((subrel R) S)->((S Y0) Z))))=> x5) as proof of ((trans S)->(((subrel R) S)->((S X0) Z)))
% Found (fun (x5:((trans S)->(((subrel R) S)->((S X0) Z)))) (x6:((trans S)->(((subrel R) S)->((S Y0) Z))))=> x5) as proof of (((trans S)->(((subrel R) S)->((S Y0) Z)))->((trans S)->(((subrel R) S)->((S X0) Z))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((S X0) Z)))) (x6:((trans S)->(((subrel R) S)->((S Y0) Z))))=> x5) as proof of (((trans S)->(((subrel R) S)->((S X0) Z)))->(((trans S)->(((subrel R) S)->((S Y0) Z)))->((trans S)->(((subrel R) S)->((S X0) Z)))))
% Found (and_rect30 (fun (x5:((trans S)->(((subrel R) S)->((S X0) Z)))) (x6:((trans S)->(((subrel R) S)->((S Y0) Z))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Z0)
% Found ((and_rect3 ((trans S)->(((subrel R) S)->((S X0) Z)))) (fun (x5:((trans S)->(((subrel R) S)->((S X0) Z)))) (x6:((trans S)->(((subrel R) S)->((S Y0) Z))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Z0)
% Found (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((S X0) Z)))->(((trans S)->(((subrel R) S)->((S Y0) Z)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((S X0) Z)))) ((trans S)->(((subrel R) S)->((S Y0) Z)))) P) x5) x4)) ((trans S)->(((subrel R) S)->((S X0) Z)))) (fun (x5:((trans S)->(((subrel R) S)->((S X0) Z)))) (x6:((trans S)->(((subrel R) S)->((S Y0) Z))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((S X0) Z)))->(((trans S)->(((subrel R) S)->((S Y0) Z)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((S X0) Z)))) ((trans S)->(((subrel R) S)->((S Y0) Z)))) P) x5) x4)) ((trans S)->(((subrel R) S)->((S X0) Z)))) (fun (x5:((trans S)->(((subrel R) S)->((S X0) Z)))) (x6:((trans S)->(((subrel R) S)->((S Y0) Z))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((S X0) Z)))->(((trans S)->(((subrel R) S)->((S Y0) Z)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((S X0) Z)))) ((trans S)->(((subrel R) S)->((S Y0) Z)))) P) x5) x4)) ((trans S)->(((subrel R) S)->((S X0) Z)))) (fun (x5:((trans S)->(((subrel R) S)->((S X0) Z)))) (x6:((trans S)->(((subrel R) S)->((S Y0) Z))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((S X0) Z)))->(((trans S)->(((subrel R) S)->((S Y0) Z)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((S X0) Z)))) ((trans S)->(((subrel R) S)->((S Y0) Z)))) P) x5) x4)) ((trans S)->(((subrel R) S)->((S X0) Z)))) (fun (x5:((trans S)->(((subrel R) S)->((S X0) Z)))) (x6:((trans S)->(((subrel R) S)->((S Y0) Z))))=> x5))) as proof of (forall (Z0:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((S X0) Z)))->(((trans S)->(((subrel R) S)->((S Y0) Z)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((S X0) Z)))) ((trans S)->(((subrel R) S)->((S Y0) Z)))) P) x5) x4)) ((trans S)->(((subrel R) S)->((S X0) Z)))) (fun (x5:((trans S)->(((subrel R) S)->((S X0) Z)))) (x6:((trans S)->(((subrel R) S)->((S Y0) Z))))=> x5))) as proof of (forall (Y:fofType) (Z0:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) Y) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((S X0) Z)))->(((trans S)->(((subrel R) S)->((S Y0) Z)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((S X0) Z)))) ((trans S)->(((subrel R) S)->((S Y0) Z)))) P) x5) x4)) ((trans S)->(((subrel R) S)->((S X0) Z)))) (fun (x5:((trans S)->(((subrel R) S)->((S X0) Z)))) (x6:((trans S)->(((subrel R) S)->((S Y0) Z))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((S x6) Z)))))
% Found x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))
% Found (fun (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))))=> x6) as proof of ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))))=> x6) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))->((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))))=> x6) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))->((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))))
% Found (and_rect20 (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Z0)
% Found ((and_rect2 ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))))=> x6))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))))=> x6))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Y0) x60))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) Z0) x60)))))=> x6))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x50) x60))))))
% Found x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))
% Found (fun (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))))=> x5) as proof of ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))))=> x5) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))->((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))))=> x5) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))->((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))))
% Found (and_rect20 (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Z0)
% Found ((and_rect2 ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) X0))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) Y0)))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> (((inv S) x500) x6))))))
% Found x6:(((subrel R) S)->((S X0) Z))
% Found (fun (x7:(((subrel R) S)->((S Y0) Z)))=> x6) as proof of (((subrel R) S)->((S X0) Z))
% Found (fun (x6:(((subrel R) S)->((S X0) Z))) (x7:(((subrel R) S)->((S Y0) Z)))=> x6) as proof of ((((subrel R) S)->((S Y0) Z))->(((subrel R) S)->((S X0) Z)))
% Found (fun (x6:(((subrel R) S)->((S X0) Z))) (x7:(((subrel R) S)->((S Y0) Z)))=> x6) as proof of ((((subrel R) S)->((S X0) Z))->((((subrel R) S)->((S Y0) Z))->(((subrel R) S)->((S X0) Z))))
% Found (and_rect30 (fun (x6:(((subrel R) S)->((S X0) Z))) (x7:(((subrel R) S)->((S Y0) Z)))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Z0)
% Found ((and_rect3 (((subrel R) S)->((S X0) Z))) (fun (x6:(((subrel R) S)->((S X0) Z))) (x7:(((subrel R) S)->((S Y0) Z)))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Z0)
% Found (((fun (P:Type) (x6:((((subrel R) S)->((S X0) Z))->((((subrel R) S)->((S Y0) Z))->P)))=> (((((and_rect (((subrel R) S)->((S X0) Z))) (((subrel R) S)->((S Y0) Z))) P) x6) x5)) (((subrel R) S)->((S X0) Z))) (fun (x6:(((subrel R) S)->((S X0) Z))) (x7:(((subrel R) S)->((S Y0) Z)))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Z0)
% Found (fun (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((S X0) Z))->((((subrel R) S)->((S Y0) Z))->P)))=> (((((and_rect (((subrel R) S)->((S X0) Z))) (((subrel R) S)->((S Y0) Z))) P) x6) x5)) (((subrel R) S)->((S X0) Z))) (fun (x6:(((subrel R) S)->((S X0) Z))) (x7:(((subrel R) S)->((S Y0) Z)))=> x6))) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Z0)
% Found (fun (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((S X0) Z))->((((subrel R) S)->((S Y0) Z))->P)))=> (((((and_rect (((subrel R) S)->((S X0) Z))) (((subrel R) S)->((S Y0) Z))) P) x6) x5)) (((subrel R) S)->((S X0) Z))) (fun (x6:(((subrel R) S)->((S X0) Z))) (x7:(((subrel R) S)->((S Y0) Z)))=> x6))) as proof of (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) Y0) Z0))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((S X0) Z))->((((subrel R) S)->((S Y0) Z))->P)))=> (((((and_rect (((subrel R) S)->((S X0) Z))) (((subrel R) S)->((S Y0) Z))) P) x6) x5)) (((subrel R) S)->((S X0) Z))) (fun (x6:(((subrel R) S)->((S X0) Z))) (x7:(((subrel R) S)->((S Y0) Z)))=> x6))) as proof of (forall (Z0:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) Y0) Z0))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((S X0) Z))->((((subrel R) S)->((S Y0) Z))->P)))=> (((((and_rect (((subrel R) S)->((S X0) Z))) (((subrel R) S)->((S Y0) Z))) P) x6) x5)) (((subrel R) S)->((S X0) Z))) (fun (x6:(((subrel R) S)->((S X0) Z))) (x7:(((subrel R) S)->((S Y0) Z)))=> x6))) as proof of (forall (Y:fofType) (Z0:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Y)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) Y) Z0))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((S X0) Z))->((((subrel R) S)->((S Y0) Z))->P)))=> (((((and_rect (((subrel R) S)->((S X0) Z))) (((subrel R) S)->((S Y0) Z))) P) x6) x5)) (((subrel R) S)->((S X0) Z))) (fun (x6:(((subrel R) S)->((S X0) Z))) (x7:(((subrel R) S)->((S Y0) Z)))=> x6))) as proof of (trans (fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S x7) Z))))
% Found x7:(((subrel R) S)->((S X) Z0))
% Found (fun (x7:(((subrel R) S)->((S X) Z0)))=> x7) as proof of (((subrel R) S)->((S X) Z0))
% Found (fun (x6:(((subrel R) S)->((S X) Y0))) (x7:(((subrel R) S)->((S X) Z0)))=> x7) as proof of ((((subrel R) S)->((S X) Z0))->(((subrel R) S)->((S X) Z0)))
% Found (fun (x6:(((subrel R) S)->((S X) Y0))) (x7:(((subrel R) S)->((S X) Z0)))=> x7) as proof of ((((subrel R) S)->((S X) Y0))->((((subrel R) S)->((S X) Z0))->(((subrel R) S)->((S X) Z0))))
% Found (and_rect30 (fun (x6:(((subrel R) S)->((S X) Y0))) (x7:(((subrel R) S)->((S X) Z0)))=> x7)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Z0)
% Found ((and_rect3 (((subrel R) S)->((S X) Z0))) (fun (x6:(((subrel R) S)->((S X) Y0))) (x7:(((subrel R) S)->((S X) Z0)))=> x7)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Z0)
% Found (((fun (P:Type) (x6:((((subrel R) S)->((S X) Y0))->((((subrel R) S)->((S X) Z0))->P)))=> (((((and_rect (((subrel R) S)->((S X) Y0))) (((subrel R) S)->((S X) Z0))) P) x6) x5)) (((subrel R) S)->((S X) Z0))) (fun (x6:(((subrel R) S)->((S X) Y0))) (x7:(((subrel R) S)->((S X) Z0)))=> x7)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Z0)
% Found (fun (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((S X) Y0))->((((subrel R) S)->((S X) Z0))->P)))=> (((((and_rect (((subrel R) S)->((S X) Y0))) (((subrel R) S)->((S X) Z0))) P) x6) x5)) (((subrel R) S)->((S X) Z0))) (fun (x6:(((subrel R) S)->((S X) Y0))) (x7:(((subrel R) S)->((S X) Z0)))=> x7))) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Z0)
% Found (fun (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((S X) Y0))->((((subrel R) S)->((S X) Z0))->P)))=> (((((and_rect (((subrel R) S)->((S X) Y0))) (((subrel R) S)->((S X) Z0))) P) x6) x5)) (((subrel R) S)->((S X) Z0))) (fun (x6:(((subrel R) S)->((S X) Y0))) (x7:(((subrel R) S)->((S X) Z0)))=> x7))) as proof of (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) Y0) Z0))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((S X) Y0))->((((subrel R) S)->((S X) Z0))->P)))=> (((((and_rect (((subrel R) S)->((S X) Y0))) (((subrel R) S)->((S X) Z0))) P) x6) x5)) (((subrel R) S)->((S X) Z0))) (fun (x6:(((subrel R) S)->((S X) Y0))) (x7:(((subrel R) S)->((S X) Z0)))=> x7))) as proof of (forall (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) Y0) Z))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((S X) Y0))->((((subrel R) S)->((S X) Z0))->P)))=> (((((and_rect (((subrel R) S)->((S X) Y0))) (((subrel R) S)->((S X) Z0))) P) x6) x5)) (((subrel R) S)->((S X) Z0))) (fun (x6:(((subrel R) S)->((S X) Y0))) (x7:(((subrel R) S)->((S X) Z0)))=> x7))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Y)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) Y) Z))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((S X) Y0))->((((subrel R) S)->((S X) Z0))->P)))=> (((((and_rect (((subrel R) S)->((S X) Y0))) (((subrel R) S)->((S X) Z0))) P) x6) x5)) (((subrel R) S)->((S X) Z0))) (fun (x6:(((subrel R) S)->((S X) Y0))) (x7:(((subrel R) S)->((S X) Z0)))=> x7))) as proof of (trans (fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((S X) x60))))
% Found x5:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))
% Found (fun (x6:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))=> x5) as proof of ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))
% Found (fun (x5:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (x6:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))=> x5) as proof of (((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))->((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))))
% Found (fun (x5:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (x6:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))=> x5) as proof of (((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))->(((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))->((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))))
% Found (and_rect20 (fun (x5:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (x6:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Z0)
% Found ((and_rect2 ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (fun (x5:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (x6:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))->(((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))) P) x5) x4)) ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (fun (x5:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (x6:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))->(((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))) P) x5) x4)) ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (fun (x5:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (x6:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))->(((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))) P) x5) x4)) ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (fun (x5:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (x6:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))->(((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))) P) x5) x4)) ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (fun (x5:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (x6:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))->(((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))) P) x5) x4)) ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (fun (x5:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (x6:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))->(((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0))))) P) x5) x4)) ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (fun (x5:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) X0))))) (x6:((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x40:fofType) (x30:fofType)=> (((inv S) x30) x6))))))
% Found x8:((S X) Z0)
% Found (fun (x8:((S X) Z0))=> x8) as proof of ((S X) Z0)
% Found (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8) as proof of (((S X) Z0)->((S X) Z0))
% Found (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8) as proof of (((S X) Y0)->(((S X) Z0)->((S X) Z0)))
% Found (and_rect30 (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8)) as proof of (((fun (x8:fofType)=> (S X)) X0) Z0)
% Found ((and_rect3 ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8)) as proof of (((fun (x8:fofType)=> (S X)) X0) Z0)
% Found (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8)) as proof of (((fun (x8:fofType)=> (S X)) X0) Z0)
% Found (fun (x6:((and (((fun (x8:fofType)=> (S X)) X0) Y0)) (((fun (x8:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (((fun (x8:fofType)=> (S X)) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x8:fofType)=> (S X)) X0) Y0)) (((fun (x8:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (((and (((fun (x8:fofType)=> (S X)) X0) Y0)) (((fun (x8:fofType)=> (S X)) Y0) Z0))->(((fun (x8:fofType)=> (S X)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType)=> (S X)) X0) Y0)) (((fun (x8:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (forall (Z:fofType), (((and (((fun (x8:fofType)=> (S X)) X0) Y0)) (((fun (x8:fofType)=> (S X)) Y0) Z))->(((fun (x8:fofType)=> (S X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType)=> (S X)) X0) Y0)) (((fun (x8:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x8:fofType)=> (S X)) X0) Y)) (((fun (x8:fofType)=> (S X)) Y) Z))->(((fun (x8:fofType)=> (S X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType)=> (S X)) X0) Y0)) (((fun (x8:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (trans (fun (x8:fofType)=> (S X)))
% Found x7:((S X0) Z)
% Found (fun (x8:((S Y0) Z))=> x7) as proof of ((S X0) Z)
% Found (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7) as proof of (((S Y0) Z)->((S X0) Z))
% Found (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7) as proof of (((S X0) Z)->(((S Y0) Z)->((S X0) Z)))
% Found (and_rect30 (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Z0)
% Found ((and_rect3 ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Z0)
% Found (((fun (P:Type) (x7:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x7) x6)) ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Z0)
% Found (fun (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x7) x6)) ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7))) as proof of (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x7) x6)) ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7))) as proof of (((and (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) Y0) Z0))->(((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x7) x6)) ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7))) as proof of (forall (Z0:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) Y0) Z0))->(((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x7) x6)) ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7))) as proof of (forall (Y:fofType) (Z0:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Y)) (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) Y) Z0))->(((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((S x8) Z)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x7) x6)) ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7))) as proof of (trans (fun (x8:fofType) (x70:fofType)=> ((S x8) Z)))
% Found x6:((trans S)->(((subrel R) S)->(((inv S) Z0) X)))
% Found (fun (x6:((trans S)->(((subrel R) S)->(((inv S) Z0) X))))=> x6) as proof of ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))
% Found (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z0) X))))=> x6) as proof of (((trans S)->(((subrel R) S)->(((inv S) Z0) X)))->((trans S)->(((subrel R) S)->(((inv S) Z0) X))))
% Found (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z0) X))))=> x6) as proof of (((trans S)->(((subrel R) S)->(((inv S) Y0) X)))->(((trans S)->(((subrel R) S)->(((inv S) Z0) X)))->((trans S)->(((subrel R) S)->(((inv S) Z0) X)))))
% Found (and_rect30 (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z0) X))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Z0)
% Found ((and_rect3 ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z0) X))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Z0)
% Found (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->(((inv S) Y0) X)))->(((trans S)->(((subrel R) S)->(((inv S) Z0) X)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))) P) x5) x4)) ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z0) X))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->(((inv S) Y0) X)))->(((trans S)->(((subrel R) S)->(((inv S) Z0) X)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))) P) x5) x4)) ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z0) X))))=> x6))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->(((inv S) Y0) X)))->(((trans S)->(((subrel R) S)->(((inv S) Z0) X)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))) P) x5) x4)) ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z0) X))))=> x6))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->(((inv S) Y0) X)))->(((trans S)->(((subrel R) S)->(((inv S) Z0) X)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))) P) x5) x4)) ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z0) X))))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->(((inv S) Y0) X)))->(((trans S)->(((subrel R) S)->(((inv S) Z0) X)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))) P) x5) x4)) ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z0) X))))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->(((inv S) Y0) X)))->(((trans S)->(((subrel R) S)->(((inv S) Z0) X)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))) P) x5) x4)) ((trans S)->(((subrel R) S)->(((inv S) Z0) X)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Y0) X)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z0) X))))=> x6))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) X)))))
% Found x5:((trans S)->(((subrel R) S)->(((inv S) Z) X0)))
% Found (fun (x6:((trans S)->(((subrel R) S)->(((inv S) Z) Y0))))=> x5) as proof of ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))
% Found (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z) Y0))))=> x5) as proof of (((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))->((trans S)->(((subrel R) S)->(((inv S) Z) X0))))
% Found (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z) Y0))))=> x5) as proof of (((trans S)->(((subrel R) S)->(((inv S) Z) X0)))->(((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))->((trans S)->(((subrel R) S)->(((inv S) Z) X0)))))
% Found (and_rect30 (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z) Y0))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Z0)
% Found ((and_rect3 ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z) Y0))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Z0)
% Found (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->(((inv S) Z) X0)))->(((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) ((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))) P) x5) x4)) ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z) Y0))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->(((inv S) Z) X0)))->(((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) ((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))) P) x5) x4)) ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z) Y0))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->(((inv S) Z) X0)))->(((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) ((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))) P) x5) x4)) ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z) Y0))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->(((inv S) Z) X0)))->(((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) ((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))) P) x5) x4)) ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z) Y0))))=> x5))) as proof of (forall (Z0:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->(((inv S) Z) X0)))->(((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) ((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))) P) x5) x4)) ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z) Y0))))=> x5))) as proof of (forall (Y:fofType) (Z0:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) Y) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->(((inv S) Z) X0)))->(((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) ((trans S)->(((subrel R) S)->(((inv S) Z) Y0)))) P) x5) x4)) ((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (fun (x5:((trans S)->(((subrel R) S)->(((inv S) Z) X0)))) (x6:((trans S)->(((subrel R) S)->(((inv S) Z) Y0))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z) x6)))))
% Found x8:((S X) Z0)
% Found (fun (x8:((S X) Z0))=> x8) as proof of ((S X) Z0)
% Found (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8) as proof of (((S X) Z0)->((S X) Z0))
% Found (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8) as proof of (((S X) Y0)->(((S X) Z0)->((S X) Z0)))
% Found (and_rect30 (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8)) as proof of (((fun (x4:fofType)=> (S X)) X0) Z0)
% Found ((and_rect3 ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8)) as proof of (((fun (x4:fofType)=> (S X)) X0) Z0)
% Found (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8)) as proof of (((fun (x4:fofType)=> (S X)) X0) Z0)
% Found (fun (x6:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (((fun (x4:fofType)=> (S X)) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0))->(((fun (x4:fofType)=> (S X)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (forall (Z:fofType), (((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z))->(((fun (x4:fofType)=> (S X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x4:fofType)=> (S X)) X0) Y)) (((fun (x4:fofType)=> (S X)) Y) Z))->(((fun (x4:fofType)=> (S X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x4:fofType)=> (S X)) X0) Y0)) (((fun (x4:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (trans (fun (x4:fofType)=> (S X)))
% Found x8:((S X) Z0)
% Found (fun (x8:((S X) Z0))=> x8) as proof of ((S X) Z0)
% Found (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8) as proof of (((S X) Z0)->((S X) Z0))
% Found (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8) as proof of (((S X) Y0)->(((S X) Z0)->((S X) Z0)))
% Found (and_rect30 (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8)) as proof of (((fun (x60:fofType)=> (S X)) X0) Z0)
% Found ((and_rect3 ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8)) as proof of (((fun (x60:fofType)=> (S X)) X0) Z0)
% Found (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8)) as proof of (((fun (x60:fofType)=> (S X)) X0) Z0)
% Found (fun (x6:((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (((fun (x60:fofType)=> (S X)) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z0))->(((fun (x6:fofType)=> (S X)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z))->(((fun (x6:fofType)=> (S X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType)=> (S X)) X0) Y)) (((fun (x6:fofType)=> (S X)) Y) Z))->(((fun (x6:fofType)=> (S X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x6:fofType)=> (S X)) X0) Y0)) (((fun (x6:fofType)=> (S X)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X) Y0)->(((S X) Z0)->P)))=> (((((and_rect ((S X) Y0)) ((S X) Z0)) P) x7) x6)) ((S X) Z0)) (fun (x7:((S X) Y0)) (x8:((S X) Z0))=> x8))) as proof of (trans (fun (x6:fofType)=> (S X)))
% Found x7:((S X0) Z)
% Found (fun (x8:((S Y0) Z))=> x7) as proof of ((S X0) Z)
% Found (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7) as proof of (((S Y0) Z)->((S X0) Z))
% Found (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7) as proof of (((S X0) Z)->(((S Y0) Z)->((S X0) Z)))
% Found (and_rect30 (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7)) as proof of (((fun (x60:fofType) (x50:fofType)=> ((S x60) Z)) X0) Z0)
% Found ((and_rect3 ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7)) as proof of (((fun (x60:fofType) (x50:fofType)=> ((S x60) Z)) X0) Z0)
% Found (((fun (P:Type) (x7:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x7) x6)) ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7)) as proof of (((fun (x60:fofType) (x50:fofType)=> ((S x60) Z)) X0) Z0)
% Found (fun (x6:((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x7) x6)) ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7))) as proof of (((fun (x60:fofType) (x50:fofType)=> ((S x60) Z)) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x7) x6)) ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x7) x6)) ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7))) as proof of (forall (Z0:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x7) x6)) ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7))) as proof of (forall (Y:fofType) (Z0:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((S x6) Z)) Y0) Z0)))=> (((fun (P:Type) (x7:(((S X0) Z)->(((S Y0) Z)->P)))=> (((((and_rect ((S X0) Z)) ((S Y0) Z)) P) x7) x6)) ((S X0) Z)) (fun (x7:((S X0) Z)) (x8:((S Y0) Z))=> x7))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((S x6) Z)))
% Found x7:(((subrel R) S)->(((inv S) Z0) X))
% Found (fun (x7:(((subrel R) S)->(((inv S) Z0) X)))=> x7) as proof of (((subrel R) S)->(((inv S) Z0) X))
% Found (fun (x6:(((subrel R) S)->(((inv S) Y0) X))) (x7:(((subrel R) S)->(((inv S) Z0) X)))=> x7) as proof of ((((subrel R) S)->(((inv S) Z0) X))->(((subrel R) S)->(((inv S) Z0) X)))
% Found (fun (x6:(((subrel R) S)->(((inv S) Y0) X))) (x7:(((subrel R) S)->(((inv S) Z0) X)))=> x7) as proof of ((((subrel R) S)->(((inv S) Y0) X))->((((subrel R) S)->(((inv S) Z0) X))->(((subrel R) S)->(((inv S) Z0) X))))
% Found (and_rect30 (fun (x6:(((subrel R) S)->(((inv S) Y0) X))) (x7:(((subrel R) S)->(((inv S) Z0) X)))=> x7)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Z0)
% Found ((and_rect3 (((subrel R) S)->(((inv S) Z0) X))) (fun (x6:(((subrel R) S)->(((inv S) Y0) X))) (x7:(((subrel R) S)->(((inv S) Z0) X)))=> x7)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Z0)
% Found (((fun (P:Type) (x6:((((subrel R) S)->(((inv S) Y0) X))->((((subrel R) S)->(((inv S) Z0) X))->P)))=> (((((and_rect (((subrel R) S)->(((inv S) Y0) X))) (((subrel R) S)->(((inv S) Z0) X))) P) x6) x5)) (((subrel R) S)->(((inv S) Z0) X))) (fun (x6:(((subrel R) S)->(((inv S) Y0) X))) (x7:(((subrel R) S)->(((inv S) Z0) X)))=> x7)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Z0)
% Found (fun (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->(((inv S) Y0) X))->((((subrel R) S)->(((inv S) Z0) X))->P)))=> (((((and_rect (((subrel R) S)->(((inv S) Y0) X))) (((subrel R) S)->(((inv S) Z0) X))) P) x6) x5)) (((subrel R) S)->(((inv S) Z0) X))) (fun (x6:(((subrel R) S)->(((inv S) Y0) X))) (x7:(((subrel R) S)->(((inv S) Z0) X)))=> x7))) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Z0)
% Found (fun (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->(((inv S) Y0) X))->((((subrel R) S)->(((inv S) Z0) X))->P)))=> (((((and_rect (((subrel R) S)->(((inv S) Y0) X))) (((subrel R) S)->(((inv S) Z0) X))) P) x6) x5)) (((subrel R) S)->(((inv S) Z0) X))) (fun (x6:(((subrel R) S)->(((inv S) Y0) X))) (x7:(((subrel R) S)->(((inv S) Z0) X)))=> x7))) as proof of (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) Y0) Z0))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->(((inv S) Y0) X))->((((subrel R) S)->(((inv S) Z0) X))->P)))=> (((((and_rect (((subrel R) S)->(((inv S) Y0) X))) (((subrel R) S)->(((inv S) Z0) X))) P) x6) x5)) (((subrel R) S)->(((inv S) Z0) X))) (fun (x6:(((subrel R) S)->(((inv S) Y0) X))) (x7:(((subrel R) S)->(((inv S) Z0) X)))=> x7))) as proof of (forall (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) Y0) Z))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->(((inv S) Y0) X))->((((subrel R) S)->(((inv S) Z0) X))->P)))=> (((((and_rect (((subrel R) S)->(((inv S) Y0) X))) (((subrel R) S)->(((inv S) Z0) X))) P) x6) x5)) (((subrel R) S)->(((inv S) Z0) X))) (fun (x6:(((subrel R) S)->(((inv S) Y0) X))) (x7:(((subrel R) S)->(((inv S) Z0) X)))=> x7))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Y)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) Y) Z))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->(((inv S) Y0) X))->((((subrel R) S)->(((inv S) Z0) X))->P)))=> (((((and_rect (((subrel R) S)->(((inv S) Y0) X))) (((subrel R) S)->(((inv S) Z0) X))) P) x6) x5)) (((subrel R) S)->(((inv S) Z0) X))) (fun (x6:(((subrel R) S)->(((inv S) Y0) X))) (x7:(((subrel R) S)->(((inv S) Z0) X)))=> x7))) as proof of (trans (fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) x60) X))))
% Found x6:(((subrel R) S)->(((inv S) Z) X0))
% Found (fun (x7:(((subrel R) S)->(((inv S) Z) Y0)))=> x6) as proof of (((subrel R) S)->(((inv S) Z) X0))
% Found (fun (x6:(((subrel R) S)->(((inv S) Z) X0))) (x7:(((subrel R) S)->(((inv S) Z) Y0)))=> x6) as proof of ((((subrel R) S)->(((inv S) Z) Y0))->(((subrel R) S)->(((inv S) Z) X0)))
% Found (fun (x6:(((subrel R) S)->(((inv S) Z) X0))) (x7:(((subrel R) S)->(((inv S) Z) Y0)))=> x6) as proof of ((((subrel R) S)->(((inv S) Z) X0))->((((subrel R) S)->(((inv S) Z) Y0))->(((subrel R) S)->(((inv S) Z) X0))))
% Found (and_rect30 (fun (x6:(((subrel R) S)->(((inv S) Z) X0))) (x7:(((subrel R) S)->(((inv S) Z) Y0)))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Z0)
% Found ((and_rect3 (((subrel R) S)->(((inv S) Z) X0))) (fun (x6:(((subrel R) S)->(((inv S) Z) X0))) (x7:(((subrel R) S)->(((inv S) Z) Y0)))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Z0)
% Found (((fun (P:Type) (x6:((((subrel R) S)->(((inv S) Z) X0))->((((subrel R) S)->(((inv S) Z) Y0))->P)))=> (((((and_rect (((subrel R) S)->(((inv S) Z) X0))) (((subrel R) S)->(((inv S) Z) Y0))) P) x6) x5)) (((subrel R) S)->(((inv S) Z) X0))) (fun (x6:(((subrel R) S)->(((inv S) Z) X0))) (x7:(((subrel R) S)->(((inv S) Z) Y0)))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Z0)
% Found (fun (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->(((inv S) Z) X0))->((((subrel R) S)->(((inv S) Z) Y0))->P)))=> (((((and_rect (((subrel R) S)->(((inv S) Z) X0))) (((subrel R) S)->(((inv S) Z) Y0))) P) x6) x5)) (((subrel R) S)->(((inv S) Z) X0))) (fun (x6:(((subrel R) S)->(((inv S) Z) X0))) (x7:(((subrel R) S)->(((inv S) Z) Y0)))=> x6))) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Z0)
% Found (fun (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->(((inv S) Z) X0))->((((subrel R) S)->(((inv S) Z) Y0))->P)))=> (((((and_rect (((subrel R) S)->(((inv S) Z) X0))) (((subrel R) S)->(((inv S) Z) Y0))) P) x6) x5)) (((subrel R) S)->(((inv S) Z) X0))) (fun (x6:(((subrel R) S)->(((inv S) Z) X0))) (x7:(((subrel R) S)->(((inv S) Z) Y0)))=> x6))) as proof of (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) Y0) Z0))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->(((inv S) Z) X0))->((((subrel R) S)->(((inv S) Z) Y0))->P)))=> (((((and_rect (((subrel R) S)->(((inv S) Z) X0))) (((subrel R) S)->(((inv S) Z) Y0))) P) x6) x5)) (((subrel R) S)->(((inv S) Z) X0))) (fun (x6:(((subrel R) S)->(((inv S) Z) X0))) (x7:(((subrel R) S)->(((inv S) Z) Y0)))=> x6))) as proof of (forall (Z0:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) Y0) Z0))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->(((inv S) Z) X0))->((((subrel R) S)->(((inv S) Z) Y0))->P)))=> (((((and_rect (((subrel R) S)->(((inv S) Z) X0))) (((subrel R) S)->(((inv S) Z) Y0))) P) x6) x5)) (((subrel R) S)->(((inv S) Z) X0))) (fun (x6:(((subrel R) S)->(((inv S) Z) X0))) (x7:(((subrel R) S)->(((inv S) Z) Y0)))=> x6))) as proof of (forall (Y:fofType) (Z0:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Y)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) Y) Z0))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->(((inv S) Z) X0))->((((subrel R) S)->(((inv S) Z) Y0))->P)))=> (((((and_rect (((subrel R) S)->(((inv S) Z) X0))) (((subrel R) S)->(((inv S) Z) Y0))) P) x6) x5)) (((subrel R) S)->(((inv S) Z) X0))) (fun (x6:(((subrel R) S)->(((inv S) Z) X0))) (x7:(((subrel R) S)->(((inv S) Z) Y0)))=> x6))) as proof of (trans (fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->(((inv S) Z) x7))))
% Found x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))
% Found (fun (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))))=> x6) as proof of ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))))=> x6) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))->((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))))=> x6) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))->((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))))
% Found (and_rect30 (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Z0)
% Found ((and_rect3 ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))))=> x6))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))))=> x6))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Y0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) Z0)))))))=> x6))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x60) x50))))))))
% Found x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))
% Found (fun (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))))=> x5) as proof of ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))))=> x5) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))->((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))))=> x5) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))->((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))))
% Found (and_rect30 (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Z0)
% Found ((and_rect3 ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S X0) x500))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S Y0) x500)))))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->((S x6) x500))))))))
% Found x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))
% Found (fun (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0)))))))=> x5) as proof of ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0)))))))=> x5) as proof of (((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))->((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0)))))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0)))))))=> x5) as proof of (((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))->((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))))
% Found (and_rect30 (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Z0)
% Found ((and_rect3 ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0)))))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0)))))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0)))))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0)))))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S X0)))) ((subrel R) (fun (x41:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S Y0)))) ((subrel R) (fun (x41:fofType)=> (S Y0)))))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType)=> (S x6)))) ((subrel R) (fun (x41:fofType)=> (S x6))))))))
% Found x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))
% Found (fun (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))))))=> x6) as proof of ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))))))=> x6) as proof of (((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))->((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))))))=> x6) as proof of (((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))->((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))))
% Found (and_rect30 (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Z0)
% Found ((and_rect3 ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))))))=> x6))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))))))=> x6))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))))))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))))))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Y0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) Z0)))))))=> x6))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> ((S x60) x50))))))))
% Found x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))
% Found (fun (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))))=> x5) as proof of ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))))=> x5) as proof of (((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))->((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0)))))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))))=> x5) as proof of (((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))->((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))))
% Found (and_rect30 (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Z0)
% Found ((and_rect3 ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType)=> (S x6)))) ((subrel R) (fun (x60:fofType)=> (S x6))))))))
% Found x8:(((inv S) Z0) X)
% Found (fun (x8:(((inv S) Z0) X))=> x8) as proof of (((inv S) Z0) X)
% Found (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8) as proof of ((((inv S) Z0) X)->(((inv S) Z0) X))
% Found (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8) as proof of ((((inv S) Y0) X)->((((inv S) Z0) X)->(((inv S) Z0) X)))
% Found (and_rect30 (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8)) as proof of (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Z0)
% Found ((and_rect3 (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8)) as proof of (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Z0)
% Found (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8)) as proof of (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Z0)
% Found (fun (x6:((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) Y0) Z0))->(((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (forall (Z:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) Y0) Z))->(((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Y)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) Y) Z))->(((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (trans (fun (x8:fofType) (x70:fofType)=> (((inv S) x70) X)))
% Found x7:(((inv S) Z) X0)
% Found (fun (x8:(((inv S) Z) Y0))=> x7) as proof of (((inv S) Z) X0)
% Found (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7) as proof of ((((inv S) Z) Y0)->(((inv S) Z) X0))
% Found (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7) as proof of ((((inv S) Z) X0)->((((inv S) Z) Y0)->(((inv S) Z) X0)))
% Found (and_rect30 (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Z0)
% Found ((and_rect3 (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Z0)
% Found (((fun (P:Type) (x7:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x7) x6)) (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Z0)
% Found (fun (x6:((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x7) x6)) (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7))) as proof of (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x7) x6)) (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7))) as proof of (((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) Y0) Z0))->(((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x7) x6)) (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7))) as proof of (forall (Z0:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) Y0) Z0))->(((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x7) x6)) (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7))) as proof of (forall (Y:fofType) (Z0:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Y)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) Y) Z0))->(((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x7) x6)) (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7))) as proof of (trans (fun (x8:fofType) (x70:fofType)=> (((inv S) Z) x8)))
% Found x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))
% Found (fun (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))))=> x7) as proof of ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))
% Found (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))))=> x7) as proof of (((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))->((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))))
% Found (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))))=> x7) as proof of (((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))->((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))))
% Found (and_rect30 (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))))=> x7)) as proof of (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Z0)
% Found ((and_rect3 ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))))=> x7)) as proof of (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Z0)
% Found (((fun (P:Type) (x6:(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))->P)))=> (((((and_rect ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))) P) x6) x5)) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))))=> x7)) as proof of (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Z0)
% Found (fun (x5:((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) Y0) Z0)))=> (((fun (P:Type) (x6:(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))->P)))=> (((((and_rect ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))) P) x6) x5)) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))))=> x7))) as proof of (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Z0)
% Found (fun (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) Y0) Z0)))=> (((fun (P:Type) (x6:(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))->P)))=> (((((and_rect ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))) P) x6) x5)) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))))=> x7))) as proof of (((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) Y0) Z0))->(((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) Y0) Z0)))=> (((fun (P:Type) (x6:(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))->P)))=> (((((and_rect ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))) P) x6) x5)) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))))=> x7))) as proof of (forall (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) Y0) Z))->(((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) Y0) Z0)))=> (((fun (P:Type) (x6:(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))->P)))=> (((((and_rect ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))) P) x6) x5)) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))))=> x7))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Y)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) Y) Z))->(((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))) Y0) Z0)))=> (((fun (P:Type) (x6:(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))->P)))=> (((((and_rect ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))) P) x6) x5)) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Y0)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) Z0))))))=> x7))) as proof of (trans (fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x70) x60)))))))
% Found x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))
% Found (fun (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))))=> x6) as proof of ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))
% Found (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))))=> x6) as proof of (((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))->((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))))
% Found (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))))=> x6) as proof of (((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))->((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))))
% Found (and_rect30 (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Z0)
% Found ((and_rect3 ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Z0)
% Found (((fun (P:Type) (x6:(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))->P)))=> (((((and_rect ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))) P) x6) x5)) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Z0)
% Found (fun (x5:((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) Y0) Z0)))=> (((fun (P:Type) (x6:(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))->P)))=> (((((and_rect ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))) P) x6) x5)) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))))=> x6))) as proof of (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Z0)
% Found (fun (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) Y0) Z0)))=> (((fun (P:Type) (x6:(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))->P)))=> (((((and_rect ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))) P) x6) x5)) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))))=> x6))) as proof of (((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) Y0) Z0))->(((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) Y0) Z0)))=> (((fun (P:Type) (x6:(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))->P)))=> (((((and_rect ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))) P) x6) x5)) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) Y0) Z))->(((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) Y0) Z0)))=> (((fun (P:Type) (x6:(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))->P)))=> (((((and_rect ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))) P) x6) x5)) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Y)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) Y) Z))->(((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))) Y0) Z0)))=> (((fun (P:Type) (x6:(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))->(((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))->P)))=> (((((and_rect ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600)))))) P) x6) x5)) ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (fun (x6:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S X0) x600)))))) (x7:((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S Y0) x600))))))=> x6))) as proof of (trans (fun (x7:fofType) (x60:fofType)=> ((and (trans (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600))))) ((subrel R) (fun (x70:fofType) (x600:fofType)=> (((subrel R) S)->((S x7) x600)))))))
% Found x5:((subrel R) (fun (x60:fofType)=> (S X0)))
% Found (fun (x6:((subrel R) (fun (x60:fofType)=> (S Y0))))=> x5) as proof of ((subrel R) (fun (x60:fofType)=> (S X0)))
% Found (fun (x5:((subrel R) (fun (x60:fofType)=> (S X0)))) (x6:((subrel R) (fun (x60:fofType)=> (S Y0))))=> x5) as proof of (((subrel R) (fun (x60:fofType)=> (S Y0)))->((subrel R) (fun (x60:fofType)=> (S X0))))
% Found (fun (x5:((subrel R) (fun (x60:fofType)=> (S X0)))) (x6:((subrel R) (fun (x60:fofType)=> (S Y0))))=> x5) as proof of (((subrel R) (fun (x60:fofType)=> (S X0)))->(((subrel R) (fun (x60:fofType)=> (S Y0)))->((subrel R) (fun (x60:fofType)=> (S X0)))))
% Found (and_rect20 (fun (x5:((subrel R) (fun (x60:fofType)=> (S X0)))) (x6:((subrel R) (fun (x60:fofType)=> (S Y0))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Z0)
% Found ((and_rect2 ((subrel R) (fun (x60:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x60:fofType)=> (S X0)))) (x6:((subrel R) (fun (x60:fofType)=> (S Y0))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Z0)
% Found (((fun (P:Type) (x5:(((subrel R) (fun (x60:fofType)=> (S X0)))->(((subrel R) (fun (x60:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x60:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x60:fofType)=> (S X0)))) (x6:((subrel R) (fun (x60:fofType)=> (S Y0))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x60:fofType)=> (S X0)))->(((subrel R) (fun (x60:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x60:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x60:fofType)=> (S X0)))) (x6:((subrel R) (fun (x60:fofType)=> (S Y0))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x60:fofType)=> (S X0)))->(((subrel R) (fun (x60:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x60:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x60:fofType)=> (S X0)))) (x6:((subrel R) (fun (x60:fofType)=> (S Y0))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x60:fofType)=> (S X0)))->(((subrel R) (fun (x60:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x60:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x60:fofType)=> (S X0)))) (x6:((subrel R) (fun (x60:fofType)=> (S Y0))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x60:fofType)=> (S X0)))->(((subrel R) (fun (x60:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x60:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x60:fofType)=> (S X0)))) (x6:((subrel R) (fun (x60:fofType)=> (S Y0))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x60:fofType)=> (S X0)))->(((subrel R) (fun (x60:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x60:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x60:fofType)=> (S X0)))) (x6:((subrel R) (fun (x60:fofType)=> (S Y0))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType)=> (S x6)))))
% Found x6:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))
% Found (fun (x6:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))=> x6) as proof of ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))
% Found (fun (x5:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) (x6:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))=> x6) as proof of (((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))->((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))
% Found (fun (x5:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) (x6:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))=> x6) as proof of (((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))->(((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))->((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))))
% Found (and_rect20 (fun (x5:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) (x6:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Z0)
% Found ((and_rect2 ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) (fun (x5:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) (x6:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Z0)
% Found (((fun (P:Type) (x5:(((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))->(((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))->P)))=> (((((and_rect ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) P) x5) x4)) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) (fun (x5:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) (x6:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))->(((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))->P)))=> (((((and_rect ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) P) x5) x4)) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) (fun (x5:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) (x6:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))=> x6))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))->(((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))->P)))=> (((((and_rect ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) P) x5) x4)) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) (fun (x5:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) (x6:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))=> x6))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))->(((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))->P)))=> (((((and_rect ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) P) x5) x4)) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) (fun (x5:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) (x6:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))->(((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))->P)))=> (((((and_rect ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) P) x5) x4)) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) (fun (x5:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) (x6:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))->(((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))->P)))=> (((((and_rect ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) P) x5) x4)) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0)))) (fun (x5:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Y0)))) (x6:((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) Z0))))=> x6))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((S x60) x50)))))
% Found x8:(((inv S) Z0) X)
% Found (fun (x8:(((inv S) Z0) X))=> x8) as proof of (((inv S) Z0) X)
% Found (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8) as proof of ((((inv S) Z0) X)->(((inv S) Z0) X))
% Found (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8) as proof of ((((inv S) Y0) X)->((((inv S) Z0) X)->(((inv S) Z0) X)))
% Found (and_rect30 (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8)) as proof of (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0)
% Found ((and_rect3 (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8)) as proof of (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0)
% Found (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8)) as proof of (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0)
% Found (fun (x6:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0))->(((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (forall (Z:fofType), (((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z))->(((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y) Z))->(((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) X0) Y0)) (((fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (trans (fun (x4:fofType) (x30:fofType)=> (((inv S) x30) X)))
% Found x8:(((inv S) Z0) X)
% Found (fun (x8:(((inv S) Z0) X))=> x8) as proof of (((inv S) Z0) X)
% Found (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8) as proof of ((((inv S) Z0) X)->(((inv S) Z0) X))
% Found (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8) as proof of ((((inv S) Y0) X)->((((inv S) Z0) X)->(((inv S) Z0) X)))
% Found (and_rect30 (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8)) as proof of (((fun (x60:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z0)
% Found ((and_rect3 (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8)) as proof of (((fun (x60:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z0)
% Found (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8)) as proof of (((fun (x60:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z0)
% Found (fun (x6:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (((fun (x60:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Y0) X)->((((inv S) Z0) X)->P)))=> (((((and_rect (((inv S) Y0) X)) (((inv S) Z0) X)) P) x7) x6)) (((inv S) Z0) X)) (fun (x7:(((inv S) Y0) X)) (x8:(((inv S) Z0) X))=> x8))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> (((inv S) x50) X)))
% Found x7:(((inv S) Z) X0)
% Found (fun (x8:(((inv S) Z) Y0))=> x7) as proof of (((inv S) Z) X0)
% Found (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7) as proof of ((((inv S) Z) Y0)->(((inv S) Z) X0))
% Found (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7) as proof of ((((inv S) Z) X0)->((((inv S) Z) Y0)->(((inv S) Z) X0)))
% Found (and_rect30 (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7)) as proof of (((fun (x60:fofType) (x50:fofType)=> (((inv S) Z) x60)) X0) Z0)
% Found ((and_rect3 (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7)) as proof of (((fun (x60:fofType) (x50:fofType)=> (((inv S) Z) x60)) X0) Z0)
% Found (((fun (P:Type) (x7:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x7) x6)) (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7)) as proof of (((fun (x60:fofType) (x50:fofType)=> (((inv S) Z) x60)) X0) Z0)
% Found (fun (x6:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x7) x6)) (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7))) as proof of (((fun (x60:fofType) (x50:fofType)=> (((inv S) Z) x60)) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x7) x6)) (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x7) x6)) (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7))) as proof of (forall (Z0:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x7) x6)) (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7))) as proof of (forall (Y:fofType) (Z0:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y) Z0))->(((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Z0)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)) Y0) Z0)))=> (((fun (P:Type) (x7:((((inv S) Z) X0)->((((inv S) Z) Y0)->P)))=> (((((and_rect (((inv S) Z) X0)) (((inv S) Z) Y0)) P) x7) x6)) (((inv S) Z) X0)) (fun (x7:(((inv S) Z) X0)) (x8:(((inv S) Z) Y0))=> x7))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> (((inv S) Z) x6)))
% Found x6:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))
% Found (fun (x7:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))))=> x6) as proof of (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))
% Found (fun (x6:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))))=> x6) as proof of ((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))->(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))))
% Found (fun (x6:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))))=> x6) as proof of ((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))->(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))))
% Found (and_rect30 (fun (x6:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Z0)
% Found ((and_rect3 (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Z0)
% Found (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Z0)
% Found (fun (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))))=> x6))) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Z0)
% Found (fun (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))))=> x6))) as proof of (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) Y0) Z0))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) Y0) Z))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Y)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) Y) Z))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))))=> x6))) as proof of (trans (fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x40:fofType)=> (S x7)))) ((subrel R) (fun (x40:fofType)=> (S x7)))))))
% Found x6:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))
% Found (fun (x7:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0))))))=> x6) as proof of (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))
% Found (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0))))))=> x6) as proof of ((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))->(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0))))))
% Found (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0))))))=> x6) as proof of ((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))->(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))))
% Found (and_rect30 (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0))))))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Z0)
% Found ((and_rect3 (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0))))))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Z0)
% Found (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0))))))=> x6)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Z0)
% Found (fun (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0))))))=> x6))) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Z0)
% Found (fun (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0))))))=> x6))) as proof of (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) Y0) Z0))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0))))))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) Y0) Z))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0))))))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Y)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) Y) Z))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S X0)))) ((subrel R) (fun (x600:fofType)=> (S X0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType)=> (S Y0)))) ((subrel R) (fun (x600:fofType)=> (S Y0))))))=> x6))) as proof of (trans (fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType)=> (S x7)))) ((subrel R) (fun (x600:fofType)=> (S x7)))))))
% Found x7:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))
% Found (fun (x7:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0))))))=> x7) as proof of (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))
% Found (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0))))))=> x7) as proof of ((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))->(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0))))))
% Found (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0))))))=> x7) as proof of ((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))->(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))))
% Found (and_rect30 (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0))))))=> x7)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Z0)
% Found ((and_rect3 (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0))))))=> x7)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Z0)
% Found (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0))))))=> x7)) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Z0)
% Found (fun (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0))))))=> x7))) as proof of (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Z0)
% Found (fun (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0))))))=> x7))) as proof of (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) Y0) Z0))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0))))))=> x7))) as proof of (forall (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) Y0) Z))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0))))))=> x7))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Y)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) Y) Z))->(((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x5:((and (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) X0) Y0)) (((fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))) Y0) Z0)))=> (((fun (P:Type) (x6:((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))->((((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))->P)))=> (((((and_rect (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))) P) x6) x5)) (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))))) (fun (x6:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Y0)))))) (x7:(((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) Z0))))))=> x7))) as proof of (trans (fun (x7:fofType) (x60:fofType)=> (((subrel R) S)->((and (trans (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))) ((subrel R) (fun (x600:fofType) (x500:fofType)=> ((S x600) x60)))))))
% Found x5:((subrel R) (fun (x40:fofType)=> (S X0)))
% Found (fun (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5) as proof of ((subrel R) (fun (x40:fofType)=> (S X0)))
% Found (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5) as proof of (((subrel R) (fun (x40:fofType)=> (S Y0)))->((subrel R) (fun (x40:fofType)=> (S X0))))
% Found (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5) as proof of (((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->((subrel R) (fun (x40:fofType)=> (S X0)))))
% Found (and_rect20 (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z0)
% Found ((and_rect2 ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z0)
% Found (((fun (P:Type) (x5:(((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))))
% Found x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))
% Found (fun (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))))=> x5) as proof of ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))))=> x5) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))->((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))))=> x5) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))->((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))))
% Found (and_rect30 (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Z0)
% Found ((and_rect3 ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) X0))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) Y0)))))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x500) x6))))))))
% Found x8:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))
% Found (fun (x8:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))))=> x8) as proof of ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))
% Found (fun (x7:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) (x8:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))))=> x8) as proof of (((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))->((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))))
% Found (fun (x7:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) (x8:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))))=> x8) as proof of (((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))->(((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))->((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))))
% Found (and_rect30 (fun (x7:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) (x8:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))))=> x8)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Z0)
% Found ((and_rect3 ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))) (fun (x7:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) (x8:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))))=> x8)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Z0)
% Found (((fun (P:Type) (x7:(((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))->(((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))) P) x7) x6)) ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))) (fun (x7:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) (x8:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))))=> x8)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Z0)
% Found (fun (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))->(((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))) P) x7) x6)) ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))) (fun (x7:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) (x8:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))))=> x8))) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))->(((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))) P) x7) x6)) ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))) (fun (x7:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) (x8:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))))=> x8))) as proof of (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) Y0) Z0))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))->(((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))) P) x7) x6)) ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))) (fun (x7:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) (x8:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))))=> x8))) as proof of (forall (Z:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) Y0) Z))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))->(((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))) P) x7) x6)) ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))) (fun (x7:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) (x8:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))))=> x8))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Y)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) Y) Z))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))->(((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))) P) x7) x6)) ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0))))) (fun (x7:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Y0))))) (x8:((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) Z0)))))=> x8))) as proof of (trans (fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType) (x700:fofType)=> ((S x80) x70)))) ((subrel R) (fun (x80:fofType) (x700:fofType)=> ((S x80) x70))))))
% Found x7:((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))
% Found (fun (x8:((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0)))))=> x7) as proof of ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))
% Found (fun (x7:((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (x8:((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0)))))=> x7) as proof of (((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))->((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0)))))
% Found (fun (x7:((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (x8:((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0)))))=> x7) as proof of (((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))->(((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))->((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))))
% Found (and_rect30 (fun (x7:((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (x8:((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0)))))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Z0)
% Found ((and_rect3 ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (x8:((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0)))))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Z0)
% Found (((fun (P:Type) (x7:(((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))->(((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) ((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (x8:((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0)))))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Z0)
% Found (fun (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))->(((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) ((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (x8:((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0)))))=> x7))) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))->(((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) ((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (x8:((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0)))))=> x7))) as proof of (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) Y0) Z0))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))->(((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) ((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (x8:((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0)))))=> x7))) as proof of (forall (Z:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) Y0) Z))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))->(((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) ((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (x8:((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0)))))=> x7))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Y)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) Y) Z))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))->(((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) ((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x80:fofType)=> (S X0)))) ((subrel R) (fun (x80:fofType)=> (S X0))))) (x8:((and (trans (fun (x80:fofType)=> (S Y0)))) ((subrel R) (fun (x80:fofType)=> (S Y0)))))=> x7))) as proof of (trans (fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x80:fofType)=> (S x8)))) ((subrel R) (fun (x80:fofType)=> (S x8))))))
% Found x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))
% Found (fun (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))))=> x6) as proof of ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))))=> x6) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))->((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))))
% Found (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))))=> x6) as proof of (((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))->((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))))
% Found (and_rect30 (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Z0)
% Found ((and_rect3 ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))))=> x6))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))))=> x6))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))->(((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))) P) x5) x4)) ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60))))))) (fun (x5:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Y0) x60))))))) (x6:((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) Z0) x60)))))))=> x6))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((and (trans (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60)))))) ((subrel R) (fun (x60:fofType) (x500:fofType)=> ((trans S)->(((subrel R) S)->(((inv S) x50) x60))))))))
% Found x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))
% Found (fun (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5) as proof of (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))
% Found (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5) as proof of ((forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))->(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00))))
% Found (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5) as proof of ((forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))->((forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))->(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))))
% Found (and_rect20 (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Z0)
% Found ((and_rect2 (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Z0)
% Found (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))->((forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) Y0) Z0)))=> (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))->((forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) Y0) Z0)))=> (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))->((forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) Y0) Z0)))=> (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))->((forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Z)))
% Found (fun (X1:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) Y0) Z0)))=> (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))->((forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y0:fofType), (((R X0) Y0)->((S x6) Y0)))) X1) Y)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y0:fofType), (((R X0) Y0)->((S x6) Y0)))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> (forall (Y0:fofType), (((R X0) Y0)->((S x6) Y0)))) X1) Z)))
% Found (fun (X1:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) Y0) Z0)))=> (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))->((forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> (forall (Y0:fofType), (((R X0) Y0)->((S x6) Y0)))))
% Found x6:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))
% Found (fun (x6:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0))))=> x6) as proof of (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))
% Found (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0))))=> x6) as proof of ((forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))->(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0))))
% Found (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0))))=> x6) as proof of ((forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))->((forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))->(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))))
% Found (and_rect20 (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) X1) Z0)
% Found ((and_rect2 (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) X1) Z0)
% Found (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))->((forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) X1) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))->((forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0))))=> x6))) as proof of (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) X1) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))->((forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0))))=> x6))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) X1) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))->((forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0))))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) X1) Z)))
% Found (fun (X1:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))->((forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0))))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y0:fofType), (((R X0) Y0)->((S X0) x50)))) X1) Y)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y0:fofType), (((R X0) Y0)->((S X0) x50)))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> (forall (Y0:fofType), (((R X0) Y0)->((S X0) x50)))) X1) Z)))
% Found (fun (X1:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S X0) x50)))) Y0) Z0)))=> (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))->((forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Y0)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S X0) Z0))))=> x6))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> (forall (Y0:fofType), (((R X0) Y0)->((S X0) x50)))))
% Found x7:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))
% Found (fun (x8:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x7) as proof of ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))
% Found (fun (x7:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x8:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x7) as proof of (((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0)))))
% Found (fun (x7:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x8:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x7) as proof of (((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))))
% Found (and_rect30 (fun (x7:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x8:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Z0)
% Found ((and_rect3 ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x8:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Z0)
% Found (((fun (P:Type) (x7:(((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) ((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x8:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Z0)
% Found (fun (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) ((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x8:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x7))) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) ((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x8:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x7))) as proof of (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) Y0) Z0))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) ((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x8:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x7))) as proof of (forall (Z:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) Y0) Z))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) ((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x8:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x7))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Y)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) Y) Z))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))->(((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) ((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S X0))))) (x8:((and (trans (fun (x40:fofType)=> (S Y0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))))=> x7))) as proof of (trans (fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x40:fofType)=> (S x8)))) ((subrel R) (fun (x40:fofType)=> (S x8))))))
% Found x8:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))
% Found (fun (x8:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))))=> x8) as proof of ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))
% Found (fun (x7:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) (x8:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))))=> x8) as proof of (((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))->((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))))
% Found (fun (x7:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) (x8:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))))=> x8) as proof of (((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))->((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))))
% Found (and_rect30 (fun (x7:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) (x8:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))))=> x8)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Z0)
% Found ((and_rect3 ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))) (fun (x7:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) (x8:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))))=> x8)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Z0)
% Found (((fun (P:Type) (x7:(((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))) P) x7) x6)) ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))) (fun (x7:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) (x8:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))))=> x8)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Z0)
% Found (fun (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))) P) x7) x6)) ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))) (fun (x7:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) (x8:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))))=> x8))) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))) P) x7) x6)) ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))) (fun (x7:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) (x8:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))))=> x8))) as proof of (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) Y0) Z0))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))) P) x7) x6)) ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))) (fun (x7:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) (x8:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))))=> x8))) as proof of (forall (Z:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) Y0) Z))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))) P) x7) x6)) ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))) (fun (x7:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) (x8:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))))=> x8))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Y)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) Y) Z))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))->(((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))) P) x7) x6)) ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0))))) (fun (x7:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Y0))))) (x8:((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) Z0)))))=> x8))) as proof of (trans (fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType) (x50:fofType)=> ((S x60) x70)))) ((subrel R) (fun (x60:fofType) (x50:fofType)=> ((S x60) x70))))))
% Found x7:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))
% Found (fun (x8:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x7) as proof of ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))
% Found (fun (x7:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x8:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x7) as proof of (((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0)))))
% Found (fun (x7:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x8:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x7) as proof of (((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))))
% Found (and_rect30 (fun (x7:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x8:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Z0)
% Found ((and_rect3 ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x8:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Z0)
% Found (((fun (P:Type) (x7:(((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) ((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x8:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x7)) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Z0)
% Found (fun (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) ((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x8:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x7))) as proof of (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Z0)
% Found (fun (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) ((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x8:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x7))) as proof of (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) Y0) Z0))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) ((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x8:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x7))) as proof of (forall (Z:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) Y0) Z))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) ((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x8:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x7))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Y)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) Y) Z))->(((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x6:((and (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) X0) Y0)) (((fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))) Y0) Z0)))=> (((fun (P:Type) (x7:(((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))->(((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))->P)))=> (((((and_rect ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) ((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0))))) P) x7) x6)) ((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (fun (x7:((and (trans (fun (x60:fofType)=> (S X0)))) ((subrel R) (fun (x60:fofType)=> (S X0))))) (x8:((and (trans (fun (x60:fofType)=> (S Y0)))) ((subrel R) (fun (x60:fofType)=> (S Y0)))))=> x7))) as proof of (trans (fun (x8:fofType) (x70:fofType)=> ((and (trans (fun (x60:fofType)=> (S x8)))) ((subrel R) (fun (x60:fofType)=> (S x8))))))
% Found x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))
% Found (fun (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))))=> x5) as proof of ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))))=> x5) as proof of (((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))->((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))))=> x5) as proof of (((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))->((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))))
% Found (and_rect30 (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Z0)
% Found ((and_rect3 ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) Y0)))))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6)))) ((subrel R) (fun (x41:fofType) (x30:fofType)=> (((inv S) x30) x6))))))))
% Found x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))
% Found (fun (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))))))=> x6) as proof of ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))))))=> x6) as proof of (((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))->((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))))))=> x6) as proof of (((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))->((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))))
% Found (and_rect30 (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Z0)
% Found ((and_rect3 ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))))))=> x6)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))))))=> x6))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))))))=> x6))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))))))=> x6))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))))))=> x6))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Y0) x60))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) Z0) x60)))))))=> x6))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x50) x60))))))))
% Found x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))
% Found (fun (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))))))=> x5) as proof of ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))))))=> x5) as proof of (((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))->((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))))))
% Found (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))))))=> x5) as proof of (((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))->((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))))
% Found (and_rect30 (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Z0)
% Found ((and_rect3 ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Z0)
% Found (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))) Y0) Z0)))=> (((fun (P:Type) (x5:(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))->(((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))->P)))=> (((((and_rect ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0))))))) P) x5) x4)) ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (fun (x5:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) X0))))))) (x6:((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) Y0)))))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((trans S)->(((subrel R) S)->((and (trans (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6)))) ((subrel R) (fun (x60:fofType) (x5000:fofType)=> (((inv S) x5000) x6))))))))
% Found x5:((subrel R) (fun (x40:fofType)=> (S X0)))
% Found (fun (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5) as proof of ((subrel R) (fun (x40:fofType)=> (S X0)))
% Found (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5) as proof of (((subrel R) (fun (x40:fofType)=> (S Y0)))->((subrel R) (fun (x40:fofType)=> (S X0))))
% Found (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5) as proof of (((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->((subrel R) (fun (x40:fofType)=> (S X0)))))
% Found (and_rect20 (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z0)
% Found ((and_rect2 ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z0)
% Found (((fun (P:Type) (x5:(((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z0))->(((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z0))
% Found (fun (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5))) as proof of (forall (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z))->(((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5))) as proof of (forall (Y:fofType) (Z:fofType), (((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y) Z))->(((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Z)))
% Found (fun (X0:fofType) (Y0:fofType) (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) X0) Y0)) (((fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))) Y0) Z0)))=> (((fun (P:Type) (x5:(((subrel R) (fun (x40:fofType)=> (S X0)))->(((subrel R) (fun (x40:fofType)=> (S Y0)))->P)))=> (((((and_rect ((subrel R) (fun (x40:fofType)=> (S X0)))) ((subrel R) (fun (x40:fofType)=> (S Y0)))) P) x5) x4)) ((subrel R) (fun (x40:fofType)=> (S X0)))) (fun (x5:((subrel R) (fun (x40:fofType)=> (S X0)))) (x6:((subrel R) (fun (x40:fofType)=> (S Y0))))=> x5))) as proof of (trans (fun (x6:fofType) (x50:fofType)=> ((subrel R) (fun (x40:fofType)=> (S x6)))))
% Found x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))
% Found (fun (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5) as proof of (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))
% Found (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5) as proof of ((forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))->(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00))))
% Found (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5) as proof of ((forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))->((forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))->(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))))
% Found (and_rect20 (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Z0)
% Found ((and_rect2 (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Z0)
% Found (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))->((forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5)) as proof of (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Z0)
% Found (fun (x4:((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) Y0) Z0)))=> (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))->((forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5))) as proof of (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Z0)
% Found (fun (Z0:fofType) (x4:((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) X1) Y0)) (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)))) Y0) Z0)))=> (((fun (P:Type) (x5:((forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))->((forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))->P)))=> (((((and_rect (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00)))) P) x5) x4)) (forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (fun (x5:(forall (Y00:fofType), (((R X0) Y00)->((S X1) Y00)))) (x6:(forall (Y00:fofType), (((R X0) Y00)->((S Y0) Y00))))=> x5))) as proof of (((and (((fun (x6:fofType) (x50:fofType)=> (forall (Y00:fofType), (((R X0) Y00)->((S x6) Y00)
% EOF
%------------------------------------------------------------------------------