%------------------------------------------------------------------------------ % File : cocATP---0.2.0 % Problem : SET624^3 : TPTP v6.1.0. Released v3.6.0. % Transfm : none % Format : tptp:raw % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % Computer : n183.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 13:30:52 EDT 2014 % Result : Theorem 149.39s % Output : Proof 149.39s % Verified : % SZS Type : None (Parsing solution fails) % Syntax : Number of formulae : 0 % Comments : %------------------------------------------------------------------------------ %----ERROR: Could not form TPTP format derivation %------------------------------------------------------------------------------ %----ORIGINAL SYSTEM OUTPUT % % Problem : SET624^3 : TPTP v6.1.0. Released v3.6.0. % % Command : python CASC.py /export/starexec/sandbox/benchmark/theBenchmark.p % % Computer : n183.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 10:25:06 CDT 2014 % % CPUTime : 149.39 % 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/SET008^0.ax, trying next directory % FOF formula (<kernel.Constant object at 0xe68ef0>, <kernel.DependentProduct object at 0xe68488>) of role type named in_decl % Using role type % Declaring in:(fofType->((fofType->Prop)->Prop)) % FOF formula (((eq (fofType->((fofType->Prop)->Prop))) in) (fun (X:fofType) (M:(fofType->Prop))=> (M X))) of role definition named in % A new definition: (((eq (fofType->((fofType->Prop)->Prop))) in) (fun (X:fofType) (M:(fofType->Prop))=> (M X))) % Defined: in:=(fun (X:fofType) (M:(fofType->Prop))=> (M X)) % FOF formula (<kernel.Constant object at 0x10c32d8>, <kernel.DependentProduct object at 0xe68050>) of role type named is_a_decl % Using role type % Declaring is_a:(fofType->((fofType->Prop)->Prop)) % FOF formula (((eq (fofType->((fofType->Prop)->Prop))) is_a) (fun (X:fofType) (M:(fofType->Prop))=> (M X))) of role definition named is_a % A new definition: (((eq (fofType->((fofType->Prop)->Prop))) is_a) (fun (X:fofType) (M:(fofType->Prop))=> (M X))) % Defined: is_a:=(fun (X:fofType) (M:(fofType->Prop))=> (M X)) % FOF formula (<kernel.Constant object at 0xe68050>, <kernel.DependentProduct object at 0xe68488>) of role type named emptyset_decl % Using role type % Declaring emptyset:(fofType->Prop) % FOF formula (((eq (fofType->Prop)) emptyset) (fun (X:fofType)=> False)) of role definition named emptyset % A new definition: (((eq (fofType->Prop)) emptyset) (fun (X:fofType)=> False)) % Defined: emptyset:=(fun (X:fofType)=> False) % FOF formula (<kernel.Constant object at 0xe68908>, <kernel.DependentProduct object at 0xe68488>) of role type named unord_pair_decl % Using role type % Declaring unord_pair:(fofType->(fofType->(fofType->Prop))) % FOF formula (((eq (fofType->(fofType->(fofType->Prop)))) unord_pair) (fun (X:fofType) (Y:fofType) (U:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) U) Y)))) of role definition named unord_pair % A new definition: (((eq (fofType->(fofType->(fofType->Prop)))) unord_pair) (fun (X:fofType) (Y:fofType) (U:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) U) Y)))) % Defined: unord_pair:=(fun (X:fofType) (Y:fofType) (U:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) U) Y))) % FOF formula (<kernel.Constant object at 0xe684d0>, <kernel.DependentProduct object at 0x10bf680>) of role type named singleton_decl % Using role type % Declaring singleton:(fofType->(fofType->Prop)) % FOF formula (((eq (fofType->(fofType->Prop))) singleton) (fun (X:fofType) (U:fofType)=> (((eq fofType) U) X))) of role definition named singleton % A new definition: (((eq (fofType->(fofType->Prop))) singleton) (fun (X:fofType) (U:fofType)=> (((eq fofType) U) X))) % Defined: singleton:=(fun (X:fofType) (U:fofType)=> (((eq fofType) U) X)) % FOF formula (<kernel.Constant object at 0xe68050>, <kernel.DependentProduct object at 0x10bfe18>) of role type named union_decl % Using role type % Declaring union:((fofType->Prop)->((fofType->Prop)->(fofType->Prop))) % FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U)))) of role definition named union % A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U)))) % Defined: union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U))) % FOF formula (<kernel.Constant object at 0xe68050>, <kernel.DependentProduct object at 0x10bf290>) of role type named excl_union_decl % Using role type % Declaring excl_union:((fofType->Prop)->((fofType->Prop)->(fofType->Prop))) % FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) excl_union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or ((and (X U)) ((Y U)->False))) ((and ((X U)->False)) (Y U))))) of role definition named excl_union % A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) excl_union) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or ((and (X U)) ((Y U)->False))) ((and ((X U)->False)) (Y U))))) % Defined: excl_union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or ((and (X U)) ((Y U)->False))) ((and ((X U)->False)) (Y U)))) % FOF formula (<kernel.Constant object at 0xe68050>, <kernel.DependentProduct object at 0x10bf638>) of role type named intersection_decl % Using role type % Declaring intersection:((fofType->Prop)->((fofType->Prop)->(fofType->Prop))) % FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) intersection) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) (Y U)))) of role definition named intersection % A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) intersection) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) (Y U)))) % Defined: intersection:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) (Y U))) % FOF formula (<kernel.Constant object at 0x10bf638>, <kernel.DependentProduct object at 0x10bf368>) of role type named setminus_decl % Using role type % Declaring setminus:((fofType->Prop)->((fofType->Prop)->(fofType->Prop))) % FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) setminus) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) ((Y U)->False)))) of role definition named setminus % A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->(fofType->Prop)))) setminus) (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) ((Y U)->False)))) % Defined: setminus:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) ((Y U)->False))) % FOF formula (<kernel.Constant object at 0x10bf368>, <kernel.DependentProduct object at 0x10bf290>) of role type named complement_decl % Using role type % Declaring complement:((fofType->Prop)->(fofType->Prop)) % FOF formula (((eq ((fofType->Prop)->(fofType->Prop))) complement) (fun (X:(fofType->Prop)) (U:fofType)=> ((X U)->False))) of role definition named complement % A new definition: (((eq ((fofType->Prop)->(fofType->Prop))) complement) (fun (X:(fofType->Prop)) (U:fofType)=> ((X U)->False))) % Defined: complement:=(fun (X:(fofType->Prop)) (U:fofType)=> ((X U)->False)) % FOF formula (<kernel.Constant object at 0x10bf440>, <kernel.DependentProduct object at 0x10bf680>) of role type named disjoint_decl % Using role type % Declaring disjoint:((fofType->Prop)->((fofType->Prop)->Prop)) % FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) disjoint) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((eq (fofType->Prop)) ((intersection X) Y)) emptyset))) of role definition named disjoint % A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) disjoint) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((eq (fofType->Prop)) ((intersection X) Y)) emptyset))) % Defined: disjoint:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((eq (fofType->Prop)) ((intersection X) Y)) emptyset)) % FOF formula (<kernel.Constant object at 0x10bf680>, <kernel.DependentProduct object at 0xe4a2d8>) of role type named subset_decl % Using role type % Declaring subset:((fofType->Prop)->((fofType->Prop)->Prop)) % FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) subset) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U))))) of role definition named subset % A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) subset) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U))))) % Defined: subset:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U)))) % FOF formula (<kernel.Constant object at 0x10bf290>, <kernel.DependentProduct object at 0xe4a4d0>) of role type named meets_decl % Using role type % Declaring meets:((fofType->Prop)->((fofType->Prop)->Prop)) % FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) meets) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))))) of role definition named meets % A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) meets) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))))) % Defined: meets:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))) % FOF formula (<kernel.Constant object at 0x10bf290>, <kernel.DependentProduct object at 0xe4a488>) of role type named misses_decl % Using role type % Declaring misses:((fofType->Prop)->((fofType->Prop)->Prop)) % FOF formula (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) misses) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))->False))) of role definition named misses % A new definition: (((eq ((fofType->Prop)->((fofType->Prop)->Prop))) misses) (fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))->False))) % Defined: misses:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))->False)) % FOF formula (forall (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop)), ((iff ((meets X) ((union Y) Z))) ((or ((meets X) Y)) ((meets X) Z)))) of role conjecture named thm % Conjecture to prove = (forall (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop)), ((iff ((meets X) ((union Y) Z))) ((or ((meets X) Y)) ((meets X) Z)))):Prop % Parameter fofType_DUMMY:fofType. % We need to prove ['(forall (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop)), ((iff ((meets X) ((union Y) Z))) ((or ((meets X) Y)) ((meets X) Z))))'] % Parameter fofType:Type. % Definition in:=(fun (X:fofType) (M:(fofType->Prop))=> (M X)):(fofType->((fofType->Prop)->Prop)). % Definition is_a:=(fun (X:fofType) (M:(fofType->Prop))=> (M X)):(fofType->((fofType->Prop)->Prop)). % Definition emptyset:=(fun (X:fofType)=> False):(fofType->Prop). % Definition unord_pair:=(fun (X:fofType) (Y:fofType) (U:fofType)=> ((or (((eq fofType) U) X)) (((eq fofType) U) Y))):(fofType->(fofType->(fofType->Prop))). % Definition singleton:=(fun (X:fofType) (U:fofType)=> (((eq fofType) U) X)):(fofType->(fofType->Prop)). % Definition union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or (X U)) (Y U))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))). % Definition excl_union:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((or ((and (X U)) ((Y U)->False))) ((and ((X U)->False)) (Y U)))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))). % Definition intersection:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) (Y U))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))). % Definition setminus:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (U:fofType)=> ((and (X U)) ((Y U)->False))):((fofType->Prop)->((fofType->Prop)->(fofType->Prop))). % Definition complement:=(fun (X:(fofType->Prop)) (U:fofType)=> ((X U)->False)):((fofType->Prop)->(fofType->Prop)). % Definition disjoint:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((eq (fofType->Prop)) ((intersection X) Y)) emptyset)):((fofType->Prop)->((fofType->Prop)->Prop)). % Definition subset:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (forall (U:fofType), ((X U)->(Y U)))):((fofType->Prop)->((fofType->Prop)->Prop)). % Definition meets:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))):((fofType->Prop)->((fofType->Prop)->Prop)). % Definition misses:=(fun (X:(fofType->Prop)) (Y:(fofType->Prop))=> (((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))->False)):((fofType->Prop)->((fofType->Prop)->Prop)). % Trying to prove (forall (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop)), ((iff ((meets X) ((union Y) Z))) ((or ((meets X) Y)) ((meets X) Z)))) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found x3 as proof of (X x0) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found x3 as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found (fun (x4:(((union Y) Z) x1))=> x3) as proof of (X x0) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((((union Y) Z) x1)->(X x0)) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((X x1)->((((union Y) Z) x1)->(X x0))) % Found (and_rect00 (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found ((and_rect0 (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found (fun (x4:(((union Y) Z) x1))=> x3) as proof of (X x0) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((((union Y) Z) x1)->(X x0)) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((X x1)->((((union Y) Z) x1)->(X x0))) % Found (and_rect00 (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found ((and_rect0 (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found x3 as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found x3 as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x2)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x2)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Y x1)):((Y x1)->((and (X x0)) (Y x1))) % Found (conj10 (Y x1)) as proof of ((Y x1)->((and (X x0)) (Y x0))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Y x1)) as proof of ((Y x1)->((and (X x0)) (Y x0))) % Found ((fun (B:Prop)=> (((conj (X x0)) B) x3)) (Y x1)) as proof of ((Y x1)->((and (X x0)) (Y x0))) % Found ((fun (B:Prop)=> (((conj (X x0)) B) x3)) (Y x1)) as proof of ((Y x1)->((and (X x0)) (Y x0))) % Found ((fun (B:Prop)=> (((conj (X x0)) B) x3)) (Y x1)) as proof of ((Y x1)->((and (X x0)) (Y x0))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x2)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x2)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Z x1)):((Z x1)->((and (X x0)) (Z x1))) % Found (conj10 (Z x1)) as proof of ((Z x1)->((and (X x0)) (Z x0))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Z x1)) as proof of ((Z x1)->((and (X x0)) (Z x0))) % Found ((fun (B:Prop)=> (((conj (X x0)) B) x3)) (Z x1)) as proof of ((Z x1)->((and (X x0)) (Z x0))) % Found ((fun (B:Prop)=> (((conj (X x0)) B) x3)) (Z x1)) as proof of ((Z x1)->((and (X x0)) (Z x0))) % Found ((fun (B:Prop)=> (((conj (X x0)) B) x3)) (Z x1)) as proof of ((Z x1)->((and (X x0)) (Z x0))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x2)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x2)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found x3 as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found x3 as proof of (X x0) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x1) % Found x3 as proof of (X x0) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x3:(X x1) % Found x3 as proof of (X x0) % Found x3:(X x0) % Found x3 as proof of (X x2) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x3:(X x0) % Found x3 as proof of (X x2) % Found x3:(X x0) % Found x3 as proof of (X x2) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x3:(X x0) % Found x3 as proof of (X x2) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x3:(X x0) % Found x3 as proof of (X x2) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x3:(X x0) % Found x3 as proof of (X x2) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found (fun (x4:(((union Y) Z) x1))=> x3) as proof of (X x0) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((((union Y) Z) x1)->(X x0)) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((X x1)->((((union Y) Z) x1)->(X x0))) % Found (and_rect00 (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found ((and_rect0 (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (fun (x2:((and (X x1)) (((union Y) Z) x1)))=> (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3))) as proof of (X x0) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found (fun (x4:(((union Y) Z) x1))=> x3) as proof of (X x0) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((((union Y) Z) x1)->(X x0)) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((X x1)->((((union Y) Z) x1)->(X x0))) % Found (and_rect00 (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found ((and_rect0 (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (fun (x2:((and (X x1)) (((union Y) Z) x1)))=> (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3))) as proof of (X x0) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found x3 as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found x3 as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x5:(Y x1) % Instantiate: x0:=x1:fofType % Found (fun (x5:(Y x1))=> x5) as proof of (Y x0) % Found (fun (x5:(Y x1))=> x5) as proof of ((Y x1)->(Y x0)) % Found x5:(Z x1) % Instantiate: x0:=x1:fofType % Found (fun (x5:(Z x1))=> x5) as proof of (Z x0) % Found (fun (x5:(Z x1))=> x5) as proof of ((Z x1)->(Z x0)) % Found x5:(Y x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Z x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Y x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Z x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Z x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Z x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Y x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Z x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Y x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Y x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Z x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Z x1) % Instantiate: x0:=x1:fofType % Found (fun (x5:(Z x1))=> x5) as proof of (Z x0) % Found (fun (x5:(Z x1))=> x5) as proof of ((Z x1)->(Z x0)) % Found x5:(Z x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Y x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Y x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Y x1) % Instantiate: x0:=x1:fofType % Found (fun (x5:(Y x1))=> x5) as proof of (Y x0) % Found (fun (x5:(Y x1))=> x5) as proof of ((Y x1)->(Y x0)) % Found x5:(Z x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Y x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Z x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Y x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Z x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Z x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Y x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Y x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Y x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Z x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found (fun (x4:(((union Y) Z) x1))=> x3) as proof of (X x0) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((((union Y) Z) x1)->(X x0)) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((X x1)->((((union Y) Z) x1)->(X x0))) % Found (and_rect00 (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found ((and_rect0 (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found (fun (x4:(((union Y) Z) x1))=> x3) as proof of (X x0) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((((union Y) Z) x1)->(X x0)) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((X x1)->((((union Y) Z) x1)->(X x0))) % Found (and_rect00 (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found ((and_rect0 (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Y x1) % Found (fun (x5:(Y x1))=> x5) as proof of (Y x0) % Found (fun (x5:(Y x1))=> x5) as proof of ((Y x1)->(Y x0)) % Found x5:(Z x1) % Found (fun (x5:(Z x1))=> x5) as proof of (Z x0) % Found (fun (x5:(Z x1))=> x5) as proof of ((Z x1)->(Z x0)) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun 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x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x2)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x2)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found (fun (x4:(((union Y) Z) x1))=> x3) as proof of (X x0) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((((union Y) Z) x1)->(X x0)) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((X x1)->((((union Y) Z) x1)->(X x0))) % Found (and_rect00 (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found ((and_rect0 (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found (fun (x4:(((union Y) Z) x1))=> x3) as proof of (X x0) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((((union Y) Z) x1)->(X x0)) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((X x1)->((((union Y) Z) x1)->(X x0))) % Found (and_rect00 (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found ((and_rect0 (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found (fun (x4:(((union Y) Z) x0))=> x3) as proof of (X x2) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((((union Y) Z) x0)->(X x2)) % Found (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3) as proof of ((X x0)->((((union Y) Z) x0)->(X x2))) % Found (and_rect00 (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found ((and_rect0 (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found (((fun (P:Type) (x3:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x3) x1)) (X x2)) (fun (x3:(X x0)) (x4:(((union Y) Z) x0))=> x3)) as proof of (X x2) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found x3 as proof of (X x0) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found x3 as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x2:(X x0) % Instantiate: x4:=x0:fofType % Found x2 as proof of (X x4) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found (fun (x4:(((union Y) Z) x1))=> x3) as proof of (X x0) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((((union Y) Z) x1)->(X x0)) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((X x1)->((((union Y) Z) x1)->(X x0))) % Found (and_rect00 (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found ((and_rect0 (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (fun (x2:((and (X x1)) (((union Y) Z) x1)))=> (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3))) as proof of (X x0) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found (fun (x4:(((union Y) Z) x1))=> x3) as proof of (X x0) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((((union Y) Z) x1)->(X x0)) % Found (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3) as proof of ((X x1)->((((union Y) Z) x1)->(X x0))) % Found (and_rect00 (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found ((and_rect0 (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3)) as proof of (X x0) % Found (fun (x2:((and (X x1)) (((union Y) Z) x1)))=> (((fun (P:Type) (x3:((X x1)->((((union Y) Z) x1)->P)))=> (((((and_rect (X x1)) (((union Y) Z) x1)) P) x3) x2)) (X x0)) (fun (x3:(X x1)) (x4:(((union Y) Z) x1))=> x3))) as proof of (X x0) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x3:(X x1) % Found x3 as proof of (X x0) % Found x3:(X x1) % Found x3 as proof of (X x0) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x3:(X x0) % Found x3 as proof of (X x2) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x3:(X x0) % Found x3 as proof of (X x2) % Found x3:(X x0) % Found x3 as proof of (X x2) % Found x3:(X x0) % Found x3 as proof of (X x2) % Found x3:(X x0) % Found x3 as proof of (X x2) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x3:(X x0) % Found x3 as proof of (X x2) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x2:(X x0) % Found x2 as proof of (X x4) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found x3 as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x1) % Instantiate: x0:=x1:fofType % Found x3 as proof of (X x0) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x0) % Instantiate: x2:=x0:fofType % Found x3 as proof of (X x2) % Found x3:(X x1) % Instantiate: x5:=x1:fofType % Found x3 as proof of (X x5) % Found x3:(X x1) % Instantiate: x5:=x1:fofType % Found x3 as proof of (X x5) % Found x4:(Z x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Z x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (((conj1 (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (ex_intro000 ((((conj (X x5)) (Z x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((Z x0)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U))))) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Z x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Z x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (((conj1 (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (ex_intro000 ((((conj (X x5)) (Z x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((Z x0)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U))))) % Found x3:(X x1) % Instantiate: x5:=x1:fofType % Found x3 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Z x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Z x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (((conj1 (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (ex_intro000 ((((conj (X x5)) (Z x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((Z x0)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U))))) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Y x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Y x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (((conj1 (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (ex_intro000 ((((conj (X x5)) (Y x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((Y x0)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Y x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Y x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (((conj1 (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (ex_intro000 ((((conj (X x5)) (Y x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((Y x0)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))) % Found x3:(X x1) % Instantiate: x5:=x1:fofType % Found x3 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Y x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Y x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (((conj1 (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (ex_intro000 ((((conj (X x5)) (Y x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((Y x0)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Y x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Y x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (((conj1 (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (ex_intro000 ((((conj (X x5)) (Y x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((Y x0)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Y x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Y x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (((conj1 (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (ex_intro000 ((((conj (X x5)) (Y x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((Y x0)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))) % Found x4:(Z x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Z x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (((conj1 (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (ex_intro000 ((((conj (X x5)) (Z x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((Z x0)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U))))) % Found x4:(Z x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Z x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (((conj1 (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (ex_intro000 ((((conj (X x5)) (Z x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((Z x0)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U))))) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Z x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Z x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (((conj1 (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (ex_intro000 ((((conj (X x5)) (Z x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((Z x0)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U))))) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Y x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Y x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (((conj1 (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (ex_intro000 ((((conj (X x5)) (Y x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((Y x0)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U))))) % Found x3:(X x1) % Instantiate: x5:=x1:fofType % Found x3 as proof of (X x5) % Found x3:(X x1) % Instantiate: x5:=x1:fofType % Found x3 as proof of (X x5) % Found x3:(X x1) % Instantiate: x5:=x1:fofType % Found x3 as proof of (X x5) % Found x3:(X x1) % Instantiate: x5:=x1:fofType % Found x3 as proof of (X x5) % Found x5:(Z x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Y x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Y x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Y x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (((conj1 (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (ex_intro000 ((((conj (X x5)) (Y x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((meets X) Y) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((Y x0)->((meets X) Y)) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Z x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Z x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (((conj1 (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (ex_intro000 ((((conj (X x5)) (Z x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((meets X) Z) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((Z x0)->((meets X) Z)) % Found x5:(Y x1) % Instantiate: x0:=x1:fofType % Found (fun (x5:(Y x1))=> x5) as proof of (Y x0) % Found (fun (x5:(Y x1))=> x5) as proof of ((Y x1)->(Y x0)) % Found x5:(Z x1) % Instantiate: x0:=x1:fofType % Found (fun (x5:(Z x1))=> x5) as proof of (Z x0) % Found (fun (x5:(Z x1))=> x5) as proof of ((Z x1)->(Z x0)) % Found x5:(Y x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Z x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Y x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Y x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Z x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Z x1) % Instantiate: x0:=x1:fofType % Found (fun (x5:(Z x1))=> x5) as proof of (Z x0) % Found (fun (x5:(Z x1))=> x5) as proof of ((Z x1)->(Z x0)) % Found x5:(Y x1) % Instantiate: x0:=x1:fofType % Found (fun (x5:(Y x1))=> x5) as proof of (Y x0) % Found (fun (x5:(Y x1))=> x5) as proof of ((Y x1)->(Y x0)) % Found x5:(Y x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Z x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Z x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Z x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Y x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Y x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Z x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Z x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Z x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (((conj1 (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (ex_intro000 ((((conj (X x5)) (Z x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((meets X) Z) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((Z x0)->((meets X) Z)) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x5:(Y x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x4:(Y x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Y x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (((conj1 (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (ex_intro000 ((((conj (X x5)) (Y x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((meets X) Y) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((Y x0)->((meets X) Y)) % Found x5:(Z x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Z x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Y x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Y x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Z x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Z x0) % Instantiate: x2:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Y x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Y x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Y x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (((conj1 (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (ex_intro000 ((((conj (X x5)) (Y x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((meets X) Y) % Found (fun (x4:(Y x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((Y x0)->((meets X) Y)) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Z x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Z x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (((conj1 (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (ex_intro000 ((((conj (X x5)) (Z x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((meets X) Z) % Found (fun (x4:(Z x0))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))) as proof of ((Z x0)->((meets X) Z)) % Found x5:(Y x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Z x0) % Instantiate: x4:=x0:fofType % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x4:(X x2) % Instantiate: x1:=x2:fofType % Found x4 as proof of (X x1) % Found x4:(X x1) % Instantiate: x3:=x1:fofType % Found x4 as proof of (X x3) % Found x4:(X x2) % Instantiate: x1:=x2:fofType % Found x4 as proof of (X x1) % Found x4:(X x1) % Instantiate: x3:=x1:fofType % Found x4 as proof of (X x3) % Found x5:(Z x1) % Found (fun (x5:(Z x1))=> x5) as proof of (Z x0) % Found (fun (x5:(Z x1))=> x5) as proof of ((Z x1)->(Z x0)) % Found x5:(Z x0) % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Y x1) % Found (fun (x5:(Y x1))=> x5) as proof of (Y x0) % Found (fun (x5:(Y x1))=> x5) as proof of ((Y x1)->(Y x0)) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Z x0) % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Z x0) % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Z x0) % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Y x1) % Found (fun (x5:(Y x1))=> x5) as proof of (Y x0) % Found (fun (x5:(Y x1))=> x5) as proof of ((Y x1)->(Y x0)) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Z x0) % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Z x0) % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Z x1) % Found (fun (x5:(Z x1))=> x5) as proof of (Z x0) % Found (fun (x5:(Z x1))=> x5) as proof of ((Z x1)->(Z x0)) % Found x5:(Z x0) % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Z x0) % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Z x0) % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Z x0) % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Z x0) % Found (fun (x5:(Z x0))=> x5) as proof of (Z x2) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x2)) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x2) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x2)) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found x5:(Z x0) % Found (fun (x5:(Z x0))=> x5) as proof of (Z x4) % Found (fun (x5:(Z x0))=> x5) as proof of ((Z x0)->(Z x4)) % Found x5:(Y x0) % Found (fun (x5:(Y x0))=> x5) as proof of (Y x4) % Found (fun (x5:(Y x0))=> x5) as proof of ((Y x0)->(Y x4)) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x2)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x2)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found conj100:=(conj10 (Z x1)):((Z x1)->((and (X x0)) (Z x1))) % Found (conj10 (Z x1)) as proof of ((Z x1)->((and (X x0)) (Z x0))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Z x1)) as proof of ((Z x1)->((and (X x0)) (Z x0))) % Found ((fun (B:Prop)=> (((conj (X x0)) B) x3)) (Z x1)) as proof of ((Z x1)->((and (X x0)) (Z x0))) % Found ((fun (B:Prop)=> (((conj (X x0)) B) x3)) (Z x1)) as proof of ((Z x1)->((and (X x0)) (Z x0))) % Found ((fun (B:Prop)=> (((conj (X x0)) B) x3)) (Z x1)) as proof of ((Z x1)->((and (X x0)) (Z x0))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x2)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Y x1)):((Y x1)->((and (X x0)) (Y x1))) % Found (conj10 (Y x1)) as proof of ((Y x1)->((and (X x0)) (Y x0))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Y x1)) as proof of ((Y x1)->((and (X x0)) (Y x0))) % Found ((fun (B:Prop)=> (((conj (X x0)) B) x3)) (Y x1)) as proof of ((Y x1)->((and (X x0)) (Y x0))) % Found ((fun (B:Prop)=> (((conj (X x0)) B) x3)) (Y x1)) as proof of ((Y x1)->((and (X x0)) (Y x0))) % Found ((fun (B:Prop)=> (((conj (X x0)) B) x3)) (Y x1)) as proof of ((Y x1)->((and (X x0)) (Y x0))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x2)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z 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((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x2)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x2)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Z x0)) as proof of ((Z x0)->((and (X x2)) (Z x2))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x2)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> ((conj1 B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found ((fun (B:Prop)=> (((conj (X x2)) B) x3)) (Y x0)) as proof of ((Y x0)->((and (X x2)) (Y x2))) % Found x4:(X x1) % Instantiate: x3:=x1:fofType % Found x4 as proof of (X x3) % Found x4:(X x2) % Instantiate: x1:=x2:fofType % Found x4 as proof of (X x1) % Found x4:(X x1) % Instantiate: x3:=x1:fofType % Found x4 as proof of (X x3) % Found x4:(X x2) % Instantiate: x1:=x2:fofType % Found x4 as proof of (X x1) % Found x4:(X x1) % Instantiate: x3:=x1:fofType % Found x4 as proof of (X x3) % Found x4:(X x1) % Instantiate: x3:=x1:fofType % Found x4 as proof of (X x3) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Z x0)):((Z x0)->((and (X x4)) (Z x0))) % Found (conj10 (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Z x0)) as proof of ((Z x0)->((and (X x4)) (Z x4))) % Found conj100:=(conj10 (Y x0)):((Y x0)->((and (X x4)) (Y x0))) % Found (conj10 (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> ((conj1 B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found ((fun (B:Prop)=> (((conj (X x4)) B) x2)) (Y x0)) as proof of ((Y x0)->((and (X x4)) (Y x4))) % Found x4:(X x2) % Instantiate: x0:=x2:fofType % Found (fun (x5:(Y x2))=> x4) as proof of (X x0) % Found (fun (x4:(X x2)) (x5:(Y x2))=> x4) as proof of ((Y x2)->(X x0)) % Found (fun (x4:(X x2)) (x5:(Y x2))=> x4) as proof of ((X x2)->((Y x2)->(X x0))) % Found (and_rect00 (fun (x4:(X x2)) (x5:(Y x2))=> x4)) as proof of (X x0) % Found ((and_rect0 (X x0)) (fun (x4:(X x2)) (x5:(Y x2))=> x4)) as proof of (X x0) % Found (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) (X x0)) (fun (x4:(X x2)) (x5:(Y x2))=> x4)) as proof of (X x0) % Found (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) (X x0)) (fun (x4:(X x2)) (x5:(Y x2))=> x4)) as proof of (X x0) % Found or_introl00:=(or_introl0 (Z x0)):((Y x2)->((or (Y x2)) (Z x0))) % Found (or_introl0 (Z x0)) as proof of ((Y x2)->(((union Y) Z) x0)) % Found ((or_introl (Y x2)) (Z x0)) as proof of ((Y x2)->(((union Y) Z) x0)) % Found (fun (x4:(X x2))=> ((or_introl (Y x2)) (Z x0))) as proof of ((Y x2)->(((union Y) Z) x0)) % Found (fun (x4:(X x2))=> ((or_introl (Y x2)) (Z x0))) as proof of ((X x2)->((Y x2)->(((union Y) Z) x0))) % Found (and_rect00 (fun (x4:(X x2))=> ((or_introl (Y x2)) (Z x0)))) as proof of (((union Y) Z) x0) % Found ((and_rect0 (((union Y) Z) x0)) (fun (x4:(X x2))=> ((or_introl (Y x2)) (Z x0)))) as proof of (((union Y) Z) x0) % Found (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) (((union Y) Z) x0)) (fun (x4:(X x2))=> ((or_introl (Y x2)) (Z x0)))) as proof of (((union Y) Z) x0) % Found (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) (((union Y) Z) x0)) (fun (x4:(X x2))=> ((or_introl (Y x2)) (Z x0)))) as proof of (((union Y) Z) x0) % Found ((conj10 (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) (X x0)) (fun (x4:(X x2)) (x5:(Y x2))=> x4))) (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) (((union Y) Z) x0)) (fun (x4:(X x2))=> ((or_introl (Y x2)) (Z x0))))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found (((conj1 (((union Y) Z) x0)) (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) (X x0)) (fun (x4:(X x2)) (x5:(Y x2))=> x4))) (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) (((union Y) Z) x0)) (fun (x4:(X x2))=> ((or_introl (Y x2)) (Z x0))))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found ((((conj (X x0)) (((union Y) Z) x0)) (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) (X x0)) (fun (x4:(X x2)) (x5:(Y x2))=> x4))) (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) (((union Y) Z) x0)) (fun (x4:(X x2))=> ((or_introl (Y x2)) (Z x0))))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found (fun (x3:((and (X x2)) (Y x2)))=> ((((conj (X x0)) (((union Y) Z) x0)) (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) (X x0)) (fun (x4:(X x2)) (x5:(Y x2))=> x4))) (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) (((union Y) Z) x0)) (fun (x4:(X x2))=> ((or_introl (Y x2)) (Z x0)))))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found x4:(X x2) % Instantiate: x0:=x2:fofType % Found (fun (x5:(Z x2))=> x4) as proof of (X x0) % Found (fun (x4:(X x2)) (x5:(Z x2))=> x4) as proof of ((Z x2)->(X x0)) % Found (fun (x4:(X x2)) (x5:(Z x2))=> x4) as proof of ((X x2)->((Z x2)->(X x0))) % Found (and_rect00 (fun (x4:(X x2)) (x5:(Z x2))=> x4)) as proof of (X x0) % Found ((and_rect0 (X x0)) (fun (x4:(X x2)) (x5:(Z x2))=> x4)) as proof of (X x0) % Found (((fun (P:Type) (x4:((X x2)->((Z x2)->P)))=> (((((and_rect (X x2)) (Z x2)) P) x4) x3)) (X x0)) (fun (x4:(X x2)) (x5:(Z x2))=> x4)) as proof of (X x0) % Found (((fun (P:Type) (x4:((X x2)->((Z x2)->P)))=> (((((and_rect (X x2)) (Z x2)) P) x4) x3)) (X x0)) (fun (x4:(X x2)) (x5:(Z x2))=> x4)) as proof of (X x0) % Found or_intror00:=(or_intror0 (Z x2)):((Z x2)->((or (Y x0)) (Z x2))) % Found (or_intror0 (Z x2)) as proof of ((Z x2)->(((union Y) Z) x0)) % Found ((or_intror (Y x0)) (Z x2)) as proof of ((Z x2)->(((union Y) Z) x0)) % Found (fun (x4:(X x2))=> ((or_intror (Y x0)) (Z x2))) as proof of ((Z x2)->(((union Y) Z) x0)) % Found (fun (x4:(X x2))=> ((or_intror (Y x0)) (Z x2))) as proof of ((X x2)->((Z x2)->(((union Y) Z) x0))) % Found (and_rect00 (fun (x4:(X x2))=> ((or_intror (Y x0)) (Z x2)))) as proof of (((union Y) Z) x0) % Found ((and_rect0 (((union Y) Z) x0)) (fun (x4:(X x2))=> ((or_intror (Y x0)) (Z x2)))) as proof of (((union Y) Z) x0) % Found (((fun (P:Type) (x4:((X x2)->((Z x2)->P)))=> (((((and_rect (X x2)) (Z x2)) P) x4) x3)) (((union Y) Z) x0)) (fun (x4:(X x2))=> ((or_intror (Y x0)) (Z x2)))) as proof of (((union Y) Z) x0) % Found (((fun (P:Type) (x4:((X x2)->((Z x2)->P)))=> (((((and_rect (X x2)) (Z x2)) P) x4) x3)) (((union Y) Z) x0)) (fun (x4:(X x2))=> ((or_intror (Y x0)) (Z x2)))) as proof of (((union Y) Z) x0) % Found ((conj10 (((fun (P:Type) (x4:((X x2)->((Z x2)->P)))=> (((((and_rect (X x2)) (Z x2)) P) x4) x3)) (X x0)) (fun (x4:(X x2)) (x5:(Z x2))=> x4))) (((fun (P:Type) (x4:((X x2)->((Z x2)->P)))=> (((((and_rect (X x2)) (Z x2)) P) x4) x3)) (((union Y) Z) x0)) (fun (x4:(X x2))=> ((or_intror (Y x0)) (Z x2))))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found (((conj1 (((union Y) Z) x0)) (((fun (P:Type) (x4:((X x2)->((Z x2)->P)))=> (((((and_rect (X x2)) (Z x2)) P) x4) x3)) (X x0)) (fun (x4:(X x2)) (x5:(Z 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Y) Z) x0)) % Found x4:(Z x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Z x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (((conj1 (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found ((((conj (X x5)) (Z x5)) x2) x4) as proof of ((and (X x5)) (Z x5)) % Found (ex_intro000 ((((conj (X x5)) (Z x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)) as proof of ((ex fofType) (fun 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(U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)))) as proof of ((or ((meets X) Y)) ((meets X) Z)) % Found (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)))) as proof of ((Z x0)->((or ((meets X) Y)) ((meets X) Z))) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x2:(X x0) % Instantiate: x5:=x0:fofType % Found x2 as proof of (X x5) % Found x4:(Y x0) % Instantiate: x5:=x0:fofType % Found x4 as proof of (Y x5) % Found ((conj10 x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (((conj1 (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found ((((conj (X x5)) (Y x5)) x2) x4) as proof of ((and (X x5)) (Y x5)) % Found (ex_intro000 ((((conj (X x5)) (Y x5)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((ex_intro00 x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)) as proof of ((meets X) Y) % Found (or_introl00 ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((or ((meets X) Y)) ((meets X) Z)) % Found ((or_introl0 ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((or ((meets X) Y)) ((meets X) Z)) % Found (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))) as proof of ((or ((meets X) Y)) ((meets X) Z)) % Found (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)))) as proof of ((or ((meets X) Y)) ((meets X) Z)) % Found (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4)))) as proof of ((Y x0)->((or ((meets X) Y)) ((meets X) Z))) % Found ((or_ind00 (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror 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x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)))))) as proof of ((((union Y) Z) x0)->((or ((meets X) Y)) ((meets X) Z))) % Found (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)))))) as proof of ((X x0)->((((union Y) Z) x0)->((or ((meets X) Y)) ((meets X) Z)))) % Found (and_rect00 (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))) as proof of ((or ((meets X) Y)) ((meets X) Z)) % Found ((and_rect0 ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj 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(x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)))))))) as proof of ((or ((meets X) Y)) ((meets X) Z)) % Found (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)))))))) as proof of (((and (X x0)) (((union Y) Z) x0))->((or ((meets X) Y)) ((meets X) Z))) % Found (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)))))))) as proof of (forall (x:fofType), (((and (X x)) (((union Y) Z) x))->((or ((meets X) Y)) ((meets X) Z)))) % Found (ex_ind00 (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))))) as proof of ((or ((meets X) Y)) ((meets X) Z)) % Found ((ex_ind0 ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))))) as proof of ((or ((meets X) Y)) ((meets X) Z)) % Found (((fun (P:Prop) (x0:(forall (x:fofType), (((and (X x)) (((union Y) Z) x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) P) x0) x)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))))) as proof of ((or ((meets X) Y)) ((meets X) Z)) % Found (fun (x:((meets X) ((union Y) Z)))=> (((fun (P:Prop) (x0:(forall (x:fofType), (((and (X x)) (((union Y) Z) x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) P) x0) x)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)))))))))) as proof of ((or ((meets X) Y)) ((meets X) Z)) % Found (fun (x:((meets X) ((union Y) Z)))=> (((fun (P:Prop) (x0:(forall (x:fofType), (((and (X x)) (((union Y) Z) x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) P) x0) x)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4)))))))))) as proof of (((meets X) ((union Y) Z))->((or ((meets X) Y)) ((meets X) Z))) % Found or_introl000:=(or_introl00 (Z x0)):((or (Y x0)) (Z x0)) % Found (or_introl00 (Z x0)) as proof of (((union Y) Z) x0) % Found ((fun (B:Prop)=> ((or_introl0 B) x5)) (Z x0)) as proof of (((union Y) Z) x0) % Found ((fun (B:Prop)=> (((or_introl (Y x0)) B) x5)) (Z x0)) as proof of (((union Y) Z) x0) % Found ((fun (B:Prop)=> (((or_introl (Y x0)) B) x5)) (Z x0)) as proof of (((union Y) Z) x0) % Found ((conj10 x4) ((fun (B:Prop)=> (((or_introl (Y x0)) B) x5)) (Z x0))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found (((conj1 (((union Y) Z) x0)) x4) ((fun (B:Prop)=> (((or_introl (Y x0)) B) x5)) (Z x0))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found ((((conj (X x0)) (((union Y) Z) x0)) x4) ((fun (B:Prop)=> (((or_introl (Y x0)) B) x5)) (Z x0))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found (fun (x5:(Y x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) ((fun (B:Prop)=> (((or_introl (Y x0)) B) x5)) (Z x0)))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found (fun (x4:(X x2)) (x5:(Y x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) ((fun (B:Prop)=> (((or_introl (Y x0)) B) x5)) (Z x0)))) as proof of ((Y x2)->((and (X x0)) (((union Y) Z) x0))) % Found (fun (x4:(X x2)) (x5:(Y x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) ((fun (B:Prop)=> (((or_introl (Y x0)) B) x5)) (Z x0)))) as proof of ((X x2)->((Y x2)->((and (X x0)) (((union Y) Z) x0)))) % Found (and_rect00 (fun (x4:(X x2)) (x5:(Y x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) ((fun (B:Prop)=> (((or_introl (Y x0)) B) x5)) (Z x0))))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found ((and_rect0 ((and (X x0)) (((union Y) Z) x0))) (fun (x4:(X x2)) (x5:(Y x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) ((fun (B:Prop)=> (((or_introl (Y x0)) B) x5)) (Z x0))))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) ((and (X x0)) (((union Y) Z) x0))) (fun (x4:(X x2)) (x5:(Y x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) ((fun (B:Prop)=> (((or_introl (Y x0)) B) x5)) (Z x0))))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found (fun (x3:((and (X x2)) (Y x2)))=> (((fun (P:Type) (x4:((X x2)->((Y x2)->P)))=> (((((and_rect (X x2)) (Y x2)) P) x4) x3)) ((and (X x0)) (((union Y) Z) x0))) (fun (x4:(X x2)) (x5:(Y x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) ((fun (B:Prop)=> (((or_introl (Y x0)) B) x5)) (Z x0)))))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found or_intror000:=(or_intror00 x5):((or (Y x0)) (Z x0)) % Found (or_intror00 x5) as proof of (((union Y) Z) x0) % Found ((or_intror0 (Z x0)) x5) as proof of (((union Y) Z) x0) % Found (((or_intror (Y x0)) (Z x0)) x5) as proof of (((union Y) Z) x0) % Found (((or_intror (Y x0)) (Z x0)) x5) as proof of (((union Y) Z) x0) % Found ((conj10 x4) (((or_intror (Y x0)) (Z x0)) x5)) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found (((conj1 (((union Y) Z) x0)) x4) (((or_intror (Y x0)) (Z x0)) x5)) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found ((((conj (X x0)) (((union Y) Z) x0)) x4) (((or_intror (Y x0)) (Z x0)) x5)) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found (fun (x5:(Z x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) (((or_intror (Y x0)) (Z x0)) x5))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found (fun (x4:(X x2)) (x5:(Z x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) (((or_intror (Y x0)) (Z x0)) x5))) as proof of ((Z x2)->((and (X x0)) (((union Y) Z) x0))) % Found (fun (x4:(X x2)) (x5:(Z x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) (((or_intror (Y x0)) (Z x0)) x5))) as proof of ((X x2)->((Z x2)->((and (X x0)) (((union Y) Z) x0)))) % Found (and_rect00 (fun (x4:(X x2)) (x5:(Z x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) (((or_intror (Y x0)) (Z x0)) x5)))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found ((and_rect0 ((and (X x0)) (((union Y) Z) x0))) (fun (x4:(X x2)) (x5:(Z x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) (((or_intror (Y x0)) (Z x0)) x5)))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found (((fun (P:Type) (x4:((X x2)->((Z x2)->P)))=> (((((and_rect (X x2)) (Z x2)) P) x4) x3)) ((and (X x0)) (((union Y) Z) x0))) (fun (x4:(X x2)) (x5:(Z x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) (((or_intror (Y x0)) (Z x0)) x5)))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found (fun (x3:((and (X x2)) (Z x2)))=> (((fun (P:Type) (x4:((X x2)->((Z x2)->P)))=> (((((and_rect (X x2)) (Z x2)) P) x4) x3)) ((and (X x0)) (((union Y) Z) x0))) (fun (x4:(X x2)) (x5:(Z x2))=> ((((conj (X x0)) (((union Y) Z) x0)) x4) (((or_intror (Y x0)) (Z x0)) x5))))) as proof of ((and (X x0)) (((union Y) Z) x0)) % Found or_introl000:=(or_introl00 (Z x5)):((or (Y x5)) (Z x5)) % Found (or_introl00 (Z x5)) as proof of (((union Y) Z) x5) % Found ((fun (B:Prop)=> ((or_introl0 B) x4)) (Z x5)) as proof of (((union Y) Z) x5) % Found ((fun (B:Prop)=> (((or_introl (Y x5)) B) x4)) (Z x5)) as proof of (((union Y) Z) x5) % Found ((fun (B:Prop)=> (((or_introl (Y x5)) B) x4)) (Z x5)) as proof of (((union Y) Z) x5) % Found ((conj10 x3) ((fun (B:Prop)=> (((or_introl (Y x5)) B) x4)) (Z x5))) as proof of ((and (X x5)) (((union Y) Z) x5)) % Found (((conj1 (((union Y) Z) x5)) x3) ((fun (B:Prop)=> (((or_introl (Y x5)) B) x4)) (Z x5))) as proof of ((and (X x5)) (((union Y) Z) x5)) % Found ((((conj (X x5)) (((union Y) Z) x5)) x3) ((fun (B:Prop)=> (((or_introl (Y x5)) B) x4)) (Z x5))) as proof of ((and (X x5)) (((union Y) Z) x5)) % Found ((((conj (X x5)) (((union Y) Z) x5)) x3) ((fun (B:Prop)=> (((or_introl (Y x5)) B) x4)) (Z x5))) as proof of ((and (X x5)) (((union Y) Z) x5)) % Found (ex_intro000 ((((conj (X x5)) (((union Y) Z) x5)) x3) ((fun (B:Prop)=> (((or_introl (Y x5)) B) x4)) (Z x5)))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found ((ex_intro00 x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (fun (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1))))) as proof of ((Y x1)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) % Found (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1))))) as proof of ((X x1)->((Y x1)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))))) % Found (and_rect00 (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found ((and_rect0 ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (fun (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1))))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1))))))) as proof of (((and (X x1)) (Y x1))->((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) % Found (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1))))))) as proof of (forall (x:fofType), (((and (X x)) (Y x))->((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))))) % Found (ex_ind00 (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found ((ex_ind0 ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (((fun (P:Prop) (x1:(forall (x:fofType), (((and (X x)) (Y x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x:fofType), (((and (X x)) (Y x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1))))))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x:fofType), (((and (X x)) (Y x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1))))))))) as proof of (((meets X) Y)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) % Found or_intror000:=(or_intror00 x4):((or (Y x5)) (Z x5)) % Found (or_intror00 x4) as proof of (((union Y) Z) x5) % Found ((or_intror0 (Z x5)) x4) as proof of (((union Y) Z) x5) % Found (((or_intror (Y x5)) (Z x5)) x4) as proof of (((union Y) Z) x5) % Found (((or_intror (Y x5)) (Z x5)) x4) as proof of (((union Y) Z) x5) % Found ((conj10 x3) (((or_intror (Y x5)) (Z x5)) x4)) as proof of ((and (X x5)) (((union Y) Z) x5)) % Found (((conj1 (((union Y) Z) x5)) x3) (((or_intror (Y x5)) (Z x5)) x4)) as proof of ((and (X x5)) (((union Y) Z) x5)) % Found ((((conj (X x5)) (((union Y) Z) x5)) x3) (((or_intror (Y x5)) (Z x5)) x4)) as proof of ((and (X x5)) (((union Y) Z) x5)) % Found ((((conj (X x5)) (((union Y) Z) x5)) x3) (((or_intror (Y x5)) (Z x5)) x4)) as proof of ((and (X x5)) (((union Y) Z) x5)) % Found (ex_intro000 ((((conj (X x5)) (((union Y) Z) x5)) x3) (((or_intror (Y x5)) (Z x5)) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found ((ex_intro00 x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (((ex_intro0 (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (fun (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))) as proof of ((Z x1)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) % Found (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))) as proof of ((X x1)->((Z x1)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))))) % Found (and_rect00 (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found ((and_rect0 ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (fun (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))))) as proof of (((and (X x1)) (Z x1))->((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) % Found (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))))) as proof of (forall (x:fofType), (((and (X x)) (Z x))->((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))))) % Found (ex_ind00 (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found ((ex_ind0 ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (((fun (P:Prop) (x1:(forall (x:fofType), (((and (X x)) (Z x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x:fofType), (((and (X x)) (Z x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x:fofType), (((and (X x)) (Z x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))))))) as proof of (((meets X) Z)->((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) % Found ((or_ind00 (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x:fofType), (((and (X x)) (Y x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x:fofType), (((and (X x)) (Z x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (((or_ind0 ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x:fofType), (((and (X x)) (Y x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x:fofType), (((and (X x)) (Z x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))))))) as proof of ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) % Found (fun (x:((or ((meets X) Y)) ((meets X) Z)))=> ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))))))))) as proof of ((meets X) ((union Y) Z)) % Found (fun (x:((or ((meets X) Y)) ((meets X) Z)))=> ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))))))))) as proof of (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z))) % Found ((conj00 (fun (x:((meets X) ((union Y) Z)))=> (((fun (P:Prop) (x0:(forall (x:fofType), (((and (X x)) (((union Y) Z) x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) P) x0) x)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))))))) (fun (x:((or ((meets X) Y)) ((meets X) Z)))=> ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))))))))) as proof of ((and (((meets X) ((union Y) Z))->((or ((meets X) Y)) ((meets X) Z)))) (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z)))) % Found (((conj0 (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z)))) (fun (x:((meets X) ((union Y) Z)))=> (((fun (P:Prop) (x0:(forall (x:fofType), (((and (X x)) (((union Y) Z) x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) P) x0) x)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))))))) (fun (x:((or ((meets X) Y)) ((meets X) Z)))=> ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))))))))) as proof of ((and (((meets X) ((union Y) Z))->((or ((meets X) Y)) ((meets X) Z)))) (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z)))) % Found ((((conj (((meets X) ((union Y) Z))->((or ((meets X) Y)) ((meets X) Z)))) (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z)))) (fun (x:((meets X) ((union Y) Z)))=> (((fun (P:Prop) (x0:(forall (x:fofType), (((and (X x)) (((union Y) Z) x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) P) x0) x)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))))))) (fun (x:((or ((meets X) Y)) ((meets X) Z)))=> ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))))))))) as proof of ((and (((meets X) ((union Y) Z))->((or ((meets X) Y)) ((meets X) Z)))) (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z)))) % Found ((((conj (((meets X) ((union Y) Z))->((or ((meets X) Y)) ((meets X) Z)))) (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z)))) (fun (x:((meets X) ((union Y) Z)))=> (((fun (P:Prop) (x0:(forall (x:fofType), (((and (X x)) (((union Y) Z) x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) P) x0) x)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))))))) (fun (x:((or ((meets X) Y)) ((meets X) Z)))=> ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4))))))))))) as proof of ((and (((meets X) ((union Y) Z))->((or ((meets X) Y)) ((meets X) Z)))) (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z)))) % Found (fun (Z:(fofType->Prop))=> ((((conj (((meets X) ((union Y) Z))->((or ((meets X) Y)) ((meets X) Z)))) (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z)))) (fun (x:((meets X) ((union Y) Z)))=> (((fun (P:Prop) (x0:(forall (x:fofType), (((and (X x)) (((union Y) Z) x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) P) x0) x)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))))))) (fun (x:((or ((meets X) Y)) ((meets X) Z)))=> ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))))))))))) as proof of ((iff ((meets X) ((union Y) Z))) ((or ((meets X) Y)) ((meets X) Z))) % Found (fun (Y:(fofType->Prop)) (Z:(fofType->Prop))=> ((((conj (((meets X) ((union Y) Z))->((or ((meets X) Y)) ((meets X) Z)))) (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z)))) (fun (x:((meets X) ((union Y) Z)))=> (((fun (P:Prop) (x0:(forall (x:fofType), (((and (X x)) (((union Y) Z) x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) P) x0) x)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))))))) (fun (x:((or ((meets X) Y)) ((meets X) Z)))=> ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))))))))))) as proof of (forall (Z:(fofType->Prop)), ((iff ((meets X) ((union Y) Z))) ((or ((meets X) Y)) ((meets X) Z)))) % Found (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop))=> ((((conj (((meets X) ((union Y) Z))->((or ((meets X) Y)) ((meets X) Z)))) (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z)))) (fun (x:((meets X) ((union Y) Z)))=> (((fun (P:Prop) (x0:(forall (x:fofType), (((and (X x)) (((union Y) Z) x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) P) x0) x)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))))))) (fun (x:((or ((meets X) Y)) ((meets X) Z)))=> ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))))))))))) as proof of (forall (Y:(fofType->Prop)) (Z:(fofType->Prop)), ((iff ((meets X) ((union Y) Z))) ((or ((meets X) Y)) ((meets X) Z)))) % Found (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop))=> ((((conj (((meets X) ((union Y) Z))->((or ((meets X) Y)) ((meets X) Z)))) (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z)))) (fun (x:((meets X) ((union Y) Z)))=> (((fun (P:Prop) (x0:(forall (x:fofType), (((and (X x)) (((union Y) Z) x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) P) x0) x)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))))))) (fun (x:((or ((meets X) Y)) ((meets X) Z)))=> ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))))))))))) as proof of (forall (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop)), ((iff ((meets X) ((union Y) Z))) ((or ((meets X) Y)) ((meets X) Z)))) % Got proof (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop))=> ((((conj (((meets X) ((union Y) Z))->((or ((meets X) Y)) ((meets X) Z)))) (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z)))) (fun (x:((meets X) ((union Y) Z)))=> (((fun (P:Prop) (x0:(forall (x:fofType), (((and (X x)) (((union Y) Z) x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) P) x0) x)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))))))) (fun (x:((or ((meets X) Y)) ((meets X) Z)))=> ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))))))))))) % Time elapsed = 147.619168s % node=15775 cost=4180.000000 depth=39 % :::::::::::::::::::::: % % SZS status Theorem for /export/starexec/sandbox/benchmark/theBenchmark.p % % SZS output start Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % (fun (X:(fofType->Prop)) (Y:(fofType->Prop)) (Z:(fofType->Prop))=> ((((conj (((meets X) ((union Y) Z))->((or ((meets X) Y)) ((meets X) Z)))) (((or ((meets X) Y)) ((meets X) Z))->((meets X) ((union Y) Z)))) (fun (x:((meets X) ((union Y) Z)))=> (((fun (P:Prop) (x0:(forall (x:fofType), (((and (X x)) (((union Y) Z) x))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) P) x0) x)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x0:fofType) (x1:((and (X x0)) (((union Y) Z) x0)))=> (((fun (P:Type) (x2:((X x0)->((((union Y) Z) x0)->P)))=> (((((and_rect (X x0)) (((union Y) Z) x0)) P) x2) x1)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x2:(X x0)) (x3:(((union Y) Z) x0))=> ((((fun (P:Prop) (x4:((Y x0)->P)) (x5:((Z x0)->P))=> ((((((or_ind (Y x0)) (Z x0)) P) x4) x5) x3)) ((or ((meets X) Y)) ((meets X) Z))) (fun (x4:(Y x0))=> (((or_introl ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) x0) ((((conj (X x0)) (Y x0)) x2) x4))))) (fun (x4:(Z x0))=> (((or_intror ((meets X) Y)) ((meets X) Z)) ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) x0) ((((conj (X x0)) (Z x0)) x2) x4))))))))))) (fun (x:((or ((meets X) Y)) ((meets X) Z)))=> ((((fun (P:Prop) (x0:(((meets X) Y)->P)) (x1:(((meets X) Z)->P))=> ((((((or_ind ((meets X) Y)) ((meets X) Z)) P) x0) x1) x)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x0:((meets X) Y))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Y x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Y U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Y x1)))=> (((fun (P:Type) (x3:((X x1)->((Y x1)->P)))=> (((((and_rect (X x1)) (Y x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Y x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) ((fun (B:Prop)=> (((or_introl (Y x1)) B) x4)) (Z x1)))))))))) (fun (x0:((meets X) Z))=> (((fun (P:Prop) (x1:(forall (x0:fofType), (((and (X x0)) (Z x0))->P)))=> (((((ex_ind fofType) (fun (U:fofType)=> ((and (X U)) (Z U)))) P) x1) x0)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x1:fofType) (x2:((and (X x1)) (Z x1)))=> (((fun (P:Type) (x3:((X x1)->((Z x1)->P)))=> (((((and_rect (X x1)) (Z x1)) P) x3) x2)) ((ex fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U))))) (fun (x3:(X x1)) (x4:(Z x1))=> ((((ex_intro fofType) (fun (U:fofType)=> ((and (X U)) (((union Y) Z) U)))) x1) ((((conj (X x1)) (((union Y) Z) x1)) x3) (((or_intror (Y x1)) (Z x1)) x4)))))))))))) % % SZS output end Proof for /export/starexec/sandbox/benchmark/theBenchmark.p % EOF %------------------------------------------------------------------------------