:: C0SP2 semantic presentation begin definitionlet "X" be ($#l1_struct_0 :::"1-sorted"::: ) ; let "y" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; func "X" :::"-->"::: "y" -> ($#m1_subset_1 :::"RealMap":::) "of" "X" equals :: C0SP2:def 1 (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ($#k7_funcop_1 :::"-->"::: ) "y"); end; :: deftheorem defines :::"-->"::: C0SP2:def 1 : (Bool "for" (Set (Var "X")) "being" ($#l1_struct_0 :::"1-sorted"::: ) (Bool "for" (Set (Var "y")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "holds" (Bool (Set (Set (Var "X")) ($#k1_c0sp2 :::"-->"::: ) (Set (Var "y"))) ($#r1_hidden :::"="::: ) (Set (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ($#k7_funcop_1 :::"-->"::: ) (Set (Var "y")))))); registrationlet "X" be ($#l1_pre_topc :::"TopSpace":::); let "y" be ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) ; cluster (Set "X" ($#k1_c0sp2 :::"-->"::: ) "y") -> ($#v1_pscomp_1 :::"continuous"::: ) ; end; theorem :: C0SP2:1 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "X")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) ) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "V")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r2_hidden :::"in"::: ) (Set (Var "V"))) & (Bool (Set (Var "V")) "is" ($#v3_rcomp_1 :::"open"::: ) )) "holds" (Bool "ex" (Set (Var "W")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "W"))) & (Bool (Set (Var "W")) "is" ($#v3_pre_topc :::"open"::: ) ) & (Bool (Set (Set (Var "f")) ($#k7_relset_1 :::".:"::: ) (Set (Var "W"))) ($#r1_tarski :::"c="::: ) (Set (Var "V"))) ")" )))) ")" ))) ; definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); func :::"ContinuousFunctions"::: "X" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) equals :: C0SP2:def 2 "{" (Set (Var "f")) where f "is" ($#v1_pscomp_1 :::"continuous"::: ) ($#m1_subset_1 :::"RealMap":::) "of" "X" : (Bool verum) "}" ; end; :: deftheorem defines :::"ContinuousFunctions"::: C0SP2:def 2 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "f")) where f "is" ($#v1_pscomp_1 :::"continuous"::: ) ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "X")) : (Bool verum) "}" )); registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X") -> ($#v3_c0sp1 :::"multiplicatively-closed"::: ) ($#v4_c0sp1 :::"additively-linearly-closed"::: ) ; end; definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); func :::"R_Algebra_of_ContinuousFunctions"::: "X" -> ($#l1_funcsdom :::"AlgebraStr"::: ) equals :: C0SP2:def 3 (Set ($#g1_funcsdom :::"AlgebraStr"::: ) "(#" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" ) "," (Set "(" ($#k2_c0sp1 :::"mult_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k1_c0sp1 :::"Add_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k5_c0sp1 :::"Mult_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k4_c0sp1 :::"One_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k3_c0sp1 :::"Zero_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "#)" ); end; :: deftheorem defines :::"R_Algebra_of_ContinuousFunctions"::: C0SP2:def 3 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k3_c0sp2 :::"R_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#g1_funcsdom :::"AlgebraStr"::: ) "(#" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k2_c0sp1 :::"mult_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k1_c0sp1 :::"Add_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k5_c0sp1 :::"Mult_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k4_c0sp1 :::"One_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k3_c0sp1 :::"Zero_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "#)" ))); theorem :: C0SP2:2 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k3_c0sp2 :::"R_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X"))) "is" ($#m2_c0sp1 :::"Subalgebra"::: ) "of" (Set ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))))) ; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k3_c0sp2 :::"R_Algebra_of_ContinuousFunctions"::: ) "X") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_funcsdom :::"strict"::: ) ; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k3_c0sp2 :::"R_Algebra_of_ContinuousFunctions"::: ) "X") -> ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#v2_funcsdom :::"vector-associative"::: ) ; end; theorem :: C0SP2:3 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "," (Set (Var "H")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set "(" ($#k3_c0sp2 :::"R_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "F"))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "G"))) & (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Var "H")))) "holds" (Bool "(" (Bool (Set (Var "H")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "G")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "holds" (Bool (Set (Set (Var "h")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" )))) ; theorem :: C0SP2:4 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "," (Set (Var "H")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set "(" ($#k3_c0sp2 :::"R_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "F"))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "G")))) "holds" (Bool "(" (Bool (Set (Var "G")) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "F")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))))) ; theorem :: C0SP2:5 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "," (Set (Var "H")) "being" ($#m1_subset_1 :::"VECTOR":::) "of" (Set "(" ($#k3_c0sp2 :::"R_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "F"))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "G"))) & (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Var "H")))) "holds" (Bool "(" (Bool (Set (Var "H")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k8_group_1 :::"*"::: ) (Set (Var "G")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) "holds" (Bool (Set (Set (Var "h")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" )))) ; theorem :: C0SP2:6 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k3_c0sp2 :::"R_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k1_c0sp2 :::"-->"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: C0SP2:7 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k1_group_1 :::"1_"::: ) (Set "(" ($#k3_c0sp2 :::"R_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k1_c0sp2 :::"-->"::: ) (Num 1)))) ; theorem :: C0SP2:8 (Bool "for" (Set (Var "A")) "being" ($#l1_funcsdom :::"Algebra":::) (Bool "for" (Set (Var "A1")) "," (Set (Var "A2")) "being" ($#m2_c0sp1 :::"Subalgebra"::: ) "of" (Set (Var "A")) "st" (Bool (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A1"))) ($#r1_tarski :::"c="::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "A2"))))) "holds" (Bool (Set (Var "A1")) "is" ($#m2_c0sp1 :::"Subalgebra"::: ) "of" (Set (Var "A2"))))) ; theorem :: C0SP2:9 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k3_c0sp2 :::"R_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X"))) "is" ($#m2_c0sp1 :::"Subalgebra"::: ) "of" (Set ($#k7_c0sp1 :::"R_Algebra_of_BoundedFunctions"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))))) ; definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::); func :::"ContinuousFunctionsNorm"::: "X" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: C0SP2:def 4 (Set (Set "(" ($#k10_c0sp1 :::"BoundedFunctionsNorm"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" )); end; :: deftheorem defines :::"ContinuousFunctionsNorm"::: C0SP2:def 4 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k4_c0sp2 :::"ContinuousFunctionsNorm"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_c0sp1 :::"BoundedFunctionsNorm"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" )))); definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::); func :::"R_Normed_Algebra_of_ContinuousFunctions"::: "X" -> ($#l1_lopban_2 :::"Normed_AlgebraStr"::: ) equals :: C0SP2:def 5 (Set ($#g1_lopban_2 :::"Normed_AlgebraStr"::: ) "(#" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" ) "," (Set "(" ($#k2_c0sp1 :::"mult_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k1_c0sp1 :::"Add_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k5_c0sp1 :::"Mult_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k4_c0sp1 :::"One_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k3_c0sp1 :::"Zero_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) "X" ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k4_c0sp2 :::"ContinuousFunctionsNorm"::: ) "X" ")" ) "#)" ); end; :: deftheorem defines :::"R_Normed_Algebra_of_ContinuousFunctions"::: C0SP2:def 5 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#g1_lopban_2 :::"Normed_AlgebraStr"::: ) "(#" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k2_c0sp1 :::"mult_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k1_c0sp1 :::"Add_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k5_c0sp1 :::"Mult_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k4_c0sp1 :::"One_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k3_c0sp1 :::"Zero_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k4_c0sp2 :::"ContinuousFunctionsNorm"::: ) (Set (Var "X")) ")" ) "#)" ))); registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) "X") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_lopban_2 :::"strict"::: ) ; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) "X") -> ($#v1_group_1 :::"unital"::: ) ; end; theorem :: C0SP2:10 (Bool "for" (Set (Var "W")) "being" ($#l1_lopban_2 :::"Normed_AlgebraStr"::: ) (Bool "for" (Set (Var "V")) "being" ($#l1_funcsdom :::"Algebra":::) "st" (Bool (Bool (Set ($#g1_funcsdom :::"AlgebraStr"::: ) "(#" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "W"))) "," (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "W"))) "," (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "W"))) "," (Set "the" ($#u1_rlvect_1 :::"Mult"::: ) "of" (Set (Var "W"))) "," (Set "the" ($#u3_struct_0 :::"OneF"::: ) "of" (Set (Var "W"))) "," (Set "the" ($#u2_struct_0 :::"ZeroF"::: ) "of" (Set (Var "W"))) "#)" ) ($#r1_hidden :::"="::: ) (Set (Var "V")))) "holds" (Bool (Set (Var "W")) "is" ($#l1_funcsdom :::"Algebra":::)))) ; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) "X") -> ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v3_group_1 :::"associative"::: ) ($#v5_group_1 :::"commutative"::: ) ($#v1_vectsp_1 :::"right-distributive"::: ) ($#v3_vectsp_1 :::"right_unital"::: ) ($#v2_funcsdom :::"vector-associative"::: ) ; end; theorem :: C0SP2:11 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "holds" (Bool (Set (Set "(" ($#k5_c0sp1 :::"Mult_"::: ) "(" (Set "(" ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) ($#k1_binop_1 :::"."::: ) "(" (Num 1) "," (Set (Var "F")) ")" ) ($#r1_hidden :::"="::: ) (Set (Var "F"))))) ; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) "X") -> ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ; end; theorem :: C0SP2:12 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set (Set (Var "X")) ($#k1_c0sp2 :::"-->"::: ) (Set ($#k6_numbers :::"0"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" )))) ; theorem :: C0SP2:13 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "F")) ($#k1_normsp_0 :::".||"::: ) )))) ; theorem :: C0SP2:14 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "st" (Bool (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" )))) "holds" (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "F")) ($#k1_normsp_0 :::".||"::: ) )))) ; theorem :: C0SP2:15 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "," (Set (Var "H")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "F"))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "G"))) & (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Var "H")))) "holds" (Bool "(" (Bool (Set (Var "H")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "G")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" )))) ; theorem :: C0SP2:16 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "F"))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "G")))) "holds" (Bool "(" (Bool (Set (Var "G")) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "F")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "a")) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" ))))) ; theorem :: C0SP2:17 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "," (Set (Var "H")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "F"))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "G"))) & (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Var "H")))) "holds" (Bool "(" (Bool (Set (Var "H")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k8_group_1 :::"*"::: ) (Set (Var "G")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" )))) ; theorem :: C0SP2:18 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "F")) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" ))) ")" & "(" (Bool (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" )))) "implies" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "F")) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "F")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k18_complex1 :::"abs"::: ) (Set (Var "a")) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "F")) ($#k1_normsp_0 :::".||"::: ) ))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "F")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "G")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "F")) ($#k1_normsp_0 :::".||"::: ) ) ($#k7_real_1 :::"+"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "G")) ($#k1_normsp_0 :::".||"::: ) ))) ")" )))) ; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) "X") -> ($#v3_normsp_0 :::"discerning"::: ) ($#v4_normsp_0 :::"reflexive"::: ) ($#v2_normsp_1 :::"RealNormSpace-like"::: ) ; end; theorem :: C0SP2:19 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "," (Set (Var "H")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "h")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "F"))) & (Bool (Set (Var "g")) ($#r1_hidden :::"="::: ) (Set (Var "G"))) & (Bool (Set (Var "h")) ($#r1_hidden :::"="::: ) (Set (Var "H")))) "holds" (Bool "(" (Bool (Set (Var "H")) ($#r1_hidden :::"="::: ) (Set (Set (Var "F")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "G")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "X")) "holds" (Bool (Set (Set (Var "h")) ($#k3_funct_2 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" (Set (Var "g")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" )))) ")" )))) ; theorem :: C0SP2:20 (Bool "for" (Set (Var "X")) "being" ($#l1_normsp_1 :::"RealBanachSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "Y")) "is" ($#v2_nfcont_1 :::"closed"::: ) ) & (Bool (Set ($#k2_relset_1 :::"rng"::: ) (Set (Var "seq"))) ($#r1_tarski :::"c="::: ) (Set (Var "Y"))) & (Bool (Set (Var "seq")) "is" ($#v1_rsspace3 :::"Cauchy_sequence_by_Norm"::: ) )) "holds" (Bool "(" (Bool (Set (Var "seq")) "is" ($#v3_normsp_1 :::"convergent"::: ) ) & (Bool (Set ($#k6_normsp_1 :::"lim"::: ) (Set (Var "seq"))) ($#r2_hidden :::"in"::: ) (Set (Var "Y"))) ")" )))) ; theorem :: C0SP2:21 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "Y")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k11_c0sp1 :::"R_Normed_Algebra_of_BoundedFunctions"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_hidden :::"="::: ) (Set ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "Y")) "is" ($#v2_nfcont_1 :::"closed"::: ) ))) ; theorem :: C0SP2:22 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "seq")) "being" ($#m1_subset_1 :::"sequence":::) "of" (Set "(" ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) (Set (Var "X")) ")" ) "st" (Bool (Bool (Set (Var "seq")) "is" ($#v1_rsspace3 :::"Cauchy_sequence_by_Norm"::: ) )) "holds" (Bool (Set (Var "seq")) "is" ($#v3_normsp_1 :::"convergent"::: ) ))) ; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) "X") -> ($#v3_lopban_1 :::"complete"::: ) ; end; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_compts_1 :::"compact"::: ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k5_c0sp2 :::"R_Normed_Algebra_of_ContinuousFunctions"::: ) "X") -> ($#v5_lopban_2 :::"Banach_Algebra-like"::: ) ; end; begin theorem :: C0SP2:23 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k13_pre_poly :::"support"::: ) (Set "(" (Set (Var "f")) ($#k3_valued_1 :::"+"::: ) (Set (Var "g")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k13_pre_poly :::"support"::: ) (Set (Var "f")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k13_pre_poly :::"support"::: ) (Set (Var "g")) ")" ))))) ; theorem :: C0SP2:24 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k13_pre_poly :::"support"::: ) (Set "(" (Set (Var "a")) ($#k26_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k13_pre_poly :::"support"::: ) (Set (Var "f"))))))) ; theorem :: C0SP2:25 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "X")) "holds" (Bool (Set ($#k13_pre_poly :::"support"::: ) (Set "(" (Set (Var "f")) ($#k20_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" )) ($#r1_tarski :::"c="::: ) (Set (Set "(" ($#k13_pre_poly :::"support"::: ) (Set (Var "f")) ")" ) ($#k2_xboole_0 :::"\/"::: ) (Set "(" ($#k13_pre_poly :::"support"::: ) (Set (Var "g")) ")" ))))) ; begin definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); func :::"C_0_Functions"::: "X" -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) equals :: C0SP2:def 6 "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"RealMap":::) "of" "X" : (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) ) & (Bool "ex" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" "X" "st" (Bool "(" (Bool (Set (Var "Y")) "is" ($#v2_compts_1 :::"compact"::: ) ) & (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" "X" "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k13_pre_poly :::"support"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y"))) ")" ) ")" )) ")" ) "}" ; end; :: deftheorem defines :::"C_0_Functions"::: C0SP2:def 6 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) "{" (Set (Var "f")) where f "is" ($#m1_subset_1 :::"RealMap":::) "of" (Set (Var "X")) : (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_pscomp_1 :::"continuous"::: ) ) & (Bool "ex" (Set (Var "Y")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool "(" (Bool (Set (Var "Y")) "is" ($#v2_compts_1 :::"compact"::: ) ) & (Bool "(" "for" (Set (Var "A")) "being" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "A")) ($#r1_hidden :::"="::: ) (Set ($#k13_pre_poly :::"support"::: ) (Set (Var "f"))))) "holds" (Bool (Set ($#k2_pre_topc :::"Cl"::: ) (Set (Var "A"))) "is" ($#m1_subset_1 :::"Subset":::) "of" (Set (Var "Y"))) ")" ) ")" )) ")" ) "}" )); theorem :: C0SP2:26 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X"))) "is" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ))) ; theorem :: C0SP2:27 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "W")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k12_funcsdom :::"RAlgebra"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) "st" (Bool (Bool (Set (Var "W")) ($#r1_hidden :::"="::: ) (Set ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "W")) "is" ($#v4_c0sp1 :::"additively-linearly-closed"::: ) ))) ; theorem :: C0SP2:28 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X"))) "is" ($#v1_rlsub_1 :::"linearly-closed"::: ) )) ; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k6_c0sp2 :::"C_0_Functions"::: ) "X") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_rlsub_1 :::"linearly-closed"::: ) ; end; definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); func :::"R_VectorSpace_of_C_0_Functions"::: "X" -> ($#l1_rlvect_1 :::"RealLinearSpace":::) equals :: C0SP2:def 7 (Set ($#g1_rlvect_1 :::"RLSStruct"::: ) "(#" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) "X" ")" ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) "X" ")" ) "," (Set "(" ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) "X" ")" ) "," (Set "(" ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) "X" ")" ) "," (Set "(" ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "#)" ); end; :: deftheorem defines :::"R_VectorSpace_of_C_0_Functions"::: C0SP2:def 7 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k7_c0sp2 :::"R_VectorSpace_of_C_0_Functions"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#g1_rlvect_1 :::"RLSStruct"::: ) "(#" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "#)" ))); theorem :: C0SP2:29 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k7_c0sp2 :::"R_VectorSpace_of_C_0_Functions"::: ) (Set (Var "X"))) "is" ($#m1_rlsub_1 :::"Subspace"::: ) "of" (Set ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X")))))) ; theorem :: C0SP2:30 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k6_c0sp1 :::"BoundedFunctions"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))))))) ; definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); func :::"C_0_FunctionsNorm"::: "X" -> ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) "X" ")" ) "," (Set ($#k1_numbers :::"REAL"::: ) ) equals :: C0SP2:def 8 (Set (Set "(" ($#k10_c0sp1 :::"BoundedFunctionsNorm"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) "X" ")" )); end; :: deftheorem defines :::"C_0_FunctionsNorm"::: C0SP2:def 8 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k8_c0sp2 :::"C_0_FunctionsNorm"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k10_c0sp1 :::"BoundedFunctionsNorm"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X")) ")" )))); definitionlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); func :::"R_Normed_Space_of_C_0_Functions"::: "X" -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_normsp_1 :::"NORMSTR"::: ) equals :: C0SP2:def 9 (Set ($#g1_normsp_1 :::"NORMSTR"::: ) "(#" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) "X" ")" ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) "X" ")" ) "," (Set "(" ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) "X" ")" ) "," (Set "(" ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) "X" ")" ) "," (Set "(" ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "X") ")" ) ")" ")" ) "," (Set "(" ($#k8_c0sp2 :::"C_0_FunctionsNorm"::: ) "X" ")" ) "#)" ); end; :: deftheorem defines :::"R_Normed_Space_of_C_0_Functions"::: C0SP2:def 9 : (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k9_c0sp2 :::"R_Normed_Space_of_C_0_Functions"::: ) (Set (Var "X"))) ($#r1_hidden :::"="::: ) (Set ($#g1_normsp_1 :::"NORMSTR"::: ) "(#" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k10_rsspace :::"Zero_"::: ) "(" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k8_rsspace :::"Add_"::: ) "(" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k9_rsspace :::"Mult_"::: ) "(" (Set "(" ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X")) ")" ) "," (Set "(" ($#k10_funcsdom :::"RealVectSpace"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "X"))) ")" ) ")" ")" ) "," (Set "(" ($#k8_c0sp2 :::"C_0_FunctionsNorm"::: ) (Set (Var "X")) ")" ) "#)" ))); registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k9_c0sp2 :::"R_Normed_Space_of_C_0_Functions"::: ) "X") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v1_normsp_1 :::"strict"::: ) ; end; theorem :: C0SP2:31 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k6_c0sp2 :::"C_0_Functions"::: ) (Set (Var "X"))))) "holds" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k2_c0sp2 :::"ContinuousFunctions"::: ) (Set (Var "X")))))) ; theorem :: C0SP2:32 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k7_c0sp2 :::"R_VectorSpace_of_C_0_Functions"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k1_c0sp2 :::"-->"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: C0SP2:33 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k9_c0sp2 :::"R_Normed_Space_of_C_0_Functions"::: ) (Set (Var "X")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set (Var "X")) ($#k1_c0sp2 :::"-->"::: ) (Set ($#k6_numbers :::"0"::: ) )))) ; theorem :: C0SP2:34 (Bool "for" (Set (Var "a")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) (Bool "for" (Set (Var "F")) "," (Set (Var "G")) "being" ($#m1_subset_1 :::"Point":::) "of" (Set "(" ($#k9_c0sp2 :::"R_Normed_Space_of_C_0_Functions"::: ) (Set (Var "X")) ")" ) "holds" (Bool "(" "(" (Bool (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "F")) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) ))) "implies" (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k9_c0sp2 :::"R_Normed_Space_of_C_0_Functions"::: ) (Set (Var "X")) ")" ))) ")" & "(" (Bool (Bool (Set (Var "F")) ($#r1_hidden :::"="::: ) (Set ($#k4_struct_0 :::"0."::: ) (Set "(" ($#k9_c0sp2 :::"R_Normed_Space_of_C_0_Functions"::: ) (Set (Var "X")) ")" )))) "implies" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "F")) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set ($#k6_numbers :::"0"::: ) )) ")" & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "a")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "F")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k18_complex1 :::"abs"::: ) (Set (Var "a")) ")" ) ($#k8_real_1 :::"*"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "F")) ($#k1_normsp_0 :::".||"::: ) ))) & (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "F")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "G")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "F")) ($#k1_normsp_0 :::".||"::: ) ) ($#k7_real_1 :::"+"::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set (Var "G")) ($#k1_normsp_0 :::".||"::: ) ))) ")" )))) ; theorem :: C0SP2:35 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k9_c0sp2 :::"R_Normed_Space_of_C_0_Functions"::: ) (Set (Var "X"))) "is" ($#v2_normsp_1 :::"RealNormSpace-like"::: ) )) ; registrationlet "X" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::); cluster (Set ($#k9_c0sp2 :::"R_Normed_Space_of_C_0_Functions"::: ) "X") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v5_rlvect_1 :::"vector-distributive"::: ) ($#v6_rlvect_1 :::"scalar-distributive"::: ) ($#v7_rlvect_1 :::"scalar-associative"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#v3_normsp_0 :::"discerning"::: ) ($#v4_normsp_0 :::"reflexive"::: ) ($#v2_normsp_1 :::"RealNormSpace-like"::: ) ; end; theorem :: C0SP2:36 (Bool "for" (Set (Var "X")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_pre_topc :::"TopSpace":::) "holds" (Bool (Set ($#k9_c0sp2 :::"R_Normed_Space_of_C_0_Functions"::: ) (Set (Var "X"))) "is" ($#l1_normsp_1 :::"RealNormSpace":::))) ;