:: VECTMETR semantic presentation begin definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) ; attr "V" is :::"convex"::: means :: VECTMETR:def 1 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "V" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" "V" "st" (Bool "(" (Bool (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" ))) & (Bool (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "z")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" ))) ")" )))); end; :: deftheorem defines :::"convex"::: VECTMETR:def 1 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "V")) "is" ($#v1_vectmetr :::"convex"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set ($#k6_numbers :::"0"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))) & (Bool (Set (Var "r")) ($#r1_xxreal_0 :::"<="::: ) (Num 1))) "holds" (Bool "ex" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "st" (Bool "(" (Bool (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "z")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" ))) & (Bool (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "z")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" (Num 1) ($#k9_real_1 :::"-"::: ) (Set (Var "r")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" ))) ")" )))) ")" )); definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) ; attr "V" is :::"internal"::: means :: VECTMETR:def 2 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "V" (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1)))) "holds" (Bool (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) ")" ) & (Bool "(" "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set (Var "F")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" )) ")" )) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) ")" ) ")" )))); end; :: deftheorem defines :::"internal"::: VECTMETR:def 2 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "V")) "is" ($#v2_vectmetr :::"internal"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "p")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) )) & (Bool (Set (Var "q")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1)))) "holds" (Bool (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) ")" ) & (Bool "(" "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set (Var "F")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" )) ")" )) "holds" (Bool (Set ($#k18_complex1 :::"abs"::: ) (Set "(" (Set "(" ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ")" ) ($#k9_real_1 :::"-"::: ) (Set "(" ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "F")) ")" ) ")" )) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "q"))) ")" ) ")" )))) ")" )); theorem :: VECTMETR:1 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrSpace":::) "st" (Bool (Bool (Set (Var "V")) "is" ($#v1_vectmetr :::"convex"::: ) )) "holds" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) (Bool "for" (Set (Var "p")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool (Bool (Set (Var "p")) ($#r1_xxreal_0 :::">"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool "ex" (Set (Var "f")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "V"))) "st" (Bool "(" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Num 1)) ($#r1_hidden :::"="::: ) (Set (Var "x"))) & (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set (Var "y"))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1)))) "holds" (Bool (Set ($#k4_metric_1 :::"dist"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" ) ($#r1_xxreal_0 :::"<"::: ) (Set (Var "p"))) ")" ) & (Bool "(" "for" (Set (Var "F")) "being" ($#m2_finseq_1 :::"FinSequence"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "st" (Bool (Bool (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k3_finseq_1 :::"len"::: ) (Set (Var "f")) ")" ) ($#k9_real_1 :::"-"::: ) (Num 1))) & (Bool "(" "for" (Set (Var "i")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "st" (Bool (Bool (Num 1) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "i"))) & (Bool (Set (Var "i")) ($#r1_xxreal_0 :::"<="::: ) (Set ($#k3_finseq_1 :::"len"::: ) (Set (Var "F"))))) "holds" (Bool (Set (Set (Var "F")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i"))) ($#r1_hidden :::"="::: ) (Set ($#k4_metric_1 :::"dist"::: ) "(" (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "i")) ")" ) "," (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set "(" (Set (Var "i")) ($#k2_nat_1 :::"+"::: ) (Num 1) ")" ) ")" ) ")" )) ")" )) "holds" (Bool (Set ($#k4_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k18_rvsum_1 :::"Sum"::: ) (Set (Var "F")))) ")" ) ")" ))))) ; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_metric_1 :::"Reflexive"::: ) ($#v7_metric_1 :::"discerning"::: ) ($#v8_metric_1 :::"symmetric"::: ) ($#v9_metric_1 :::"triangle"::: ) ($#v1_vectmetr :::"convex"::: ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v2_vectmetr :::"internal"::: ) for ($#l1_metric_1 :::"MetrStruct"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_metric_1 :::"Reflexive"::: ) ($#v7_metric_1 :::"discerning"::: ) ($#v8_metric_1 :::"symmetric"::: ) ($#v9_metric_1 :::"triangle"::: ) ($#v11_metric_1 :::"Discerning"::: ) ($#v1_vectmetr :::"convex"::: ) for ($#l1_metric_1 :::"MetrStruct"::: ) ; end; begin definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) ; let "f" be ($#m1_subset_1 :::"Function":::) "of" (Set (Const "V")) "," (Set (Const "V")); attr "f" is :::"isometric"::: means :: VECTMETR:def 3 (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" "V" "holds" (Bool (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set "(" "f" ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) "," (Set "(" "f" ($#k3_funct_2 :::"."::: ) (Set (Var "y")) ")" ) ")" ))); end; :: deftheorem defines :::"isometric"::: VECTMETR:def 3 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "V")) "," (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v3_vectmetr :::"isometric"::: ) ) "iff" (Bool "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "x")) ")" ) "," (Set "(" (Set (Var "f")) ($#k3_funct_2 :::"."::: ) (Set (Var "y")) ")" ) ")" ))) ")" ))); registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) ; cluster (Set ($#k3_struct_0 :::"id"::: ) "V") -> ($#v2_funct_2 :::"onto"::: ) ($#v3_vectmetr :::"isometric"::: ) ; end; theorem :: VECTMETR:2 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) "holds" (Bool "(" (Bool (Set ($#k3_struct_0 :::"id"::: ) (Set (Var "V"))) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set ($#k3_struct_0 :::"id"::: ) (Set (Var "V"))) "is" ($#v3_vectmetr :::"isometric"::: ) ) ")" )) ; registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) ; cluster ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#v1_relat_1 :::"Relation-like"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") ($#v4_relat_1 :::"-defined"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") ($#v5_relat_1 :::"-valued"::: ) ($#v1_funct_1 :::"Function-like"::: ) ($#v1_partfun1 :::"total"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v2_funct_2 :::"onto"::: ) ($#v3_vectmetr :::"isometric"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") ($#k2_zfmisc_1 :::":]"::: ) )); end; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) ; func :::"ISOM"::: "V" -> ($#m1_hidden :::"set"::: ) means :: VECTMETR:def 4 (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) it) "iff" (Bool (Set (Var "x")) "is" ($#v2_funct_2 :::"onto"::: ) ($#v3_vectmetr :::"isometric"::: ) ($#m1_subset_1 :::"Function":::) "of" "V" "," "V") ")" )); end; :: deftheorem defines :::"ISOM"::: VECTMETR:def 4 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) (Bool "for" (Set (Var "b2")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k1_vectmetr :::"ISOM"::: ) (Set (Var "V")))) "iff" (Bool "for" (Set (Var "x")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Var "b2"))) "iff" (Bool (Set (Var "x")) "is" ($#v2_funct_2 :::"onto"::: ) ($#v3_vectmetr :::"isometric"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "V")) "," (Set (Var "V"))) ")" )) ")" ))); definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) ; :: original: :::"ISOM"::: redefine func :::"ISOM"::: "V" -> ($#m1_subset_1 :::"Subset":::) "of" (Set "(" ($#k9_funct_2 :::"Funcs"::: ) "(" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") ")" ")" ); end; registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_metric_1 :::"Reflexive"::: ) ($#v7_metric_1 :::"discerning"::: ) ($#l1_metric_1 :::"MetrStruct"::: ) ; cluster ($#v1_funct_1 :::"Function-like"::: ) ($#v1_funct_2 :::"quasi_total"::: ) ($#v3_vectmetr :::"isometric"::: ) -> ($#v2_funct_1 :::"one-to-one"::: ) for ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_zfmisc_1 :::"bool"::: ) (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") "," (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "V") ($#k2_zfmisc_1 :::":]"::: ) )); end; theorem :: VECTMETR:3 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_metric_1 :::"Reflexive"::: ) ($#v7_metric_1 :::"discerning"::: ) ($#l1_metric_1 :::"MetrStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set (Var "V")) "," (Set (Var "V")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v3_vectmetr :::"isometric"::: ) )) "holds" (Bool (Set (Var "f")) "is" ($#v2_funct_1 :::"one-to-one"::: ) ))) ; registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_metric_1 :::"Reflexive"::: ) ($#v7_metric_1 :::"discerning"::: ) ($#l1_metric_1 :::"MetrStruct"::: ) ; let "f" be ($#v2_funct_2 :::"onto"::: ) ($#v3_vectmetr :::"isometric"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Const "V")) "," (Set (Const "V")); cluster (Set "f" ($#k2_tops_2 :::"""::: ) ) -> ($#v2_funct_2 :::"onto"::: ) ($#v3_vectmetr :::"isometric"::: ) ; end; theorem :: VECTMETR:4 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v6_metric_1 :::"Reflexive"::: ) ($#v7_metric_1 :::"discerning"::: ) ($#l1_metric_1 :::"MetrStruct"::: ) (Bool "for" (Set (Var "f")) "being" ($#v2_funct_2 :::"onto"::: ) ($#v3_vectmetr :::"isometric"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "V")) "," (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_tops_2 :::"""::: ) ) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_tops_2 :::"""::: ) ) "is" ($#v3_vectmetr :::"isometric"::: ) ) ")" ))) ; registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) ; let "f", "g" be ($#v2_funct_2 :::"onto"::: ) ($#v3_vectmetr :::"isometric"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Const "V")) "," (Set (Const "V")); cluster (Set "g" ($#k3_relat_1 :::"*"::: ) "f") -> ($#v2_funct_2 :::"onto"::: ) ($#v3_vectmetr :::"isometric"::: ) for ($#m1_subset_1 :::"Function":::) "of" "V" "," "V"; end; theorem :: VECTMETR:5 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#v2_funct_2 :::"onto"::: ) ($#v3_vectmetr :::"isometric"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set (Var "V")) "," (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set (Var "g"))) "is" ($#v2_funct_2 :::"onto"::: ) ) & (Bool (Set (Set (Var "f")) ($#k1_partfun1 :::"*"::: ) (Set (Var "g"))) "is" ($#v3_vectmetr :::"isometric"::: ) ) ")" ))) ; registrationlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_metric_1 :::"MetrStruct"::: ) ; cluster (Set ($#k1_vectmetr :::"ISOM"::: ) "V") -> ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ; end; begin definitionattr "c1" is :::"strict"::: ; struct :::"RLSMetrStruct"::: -> ($#l1_rlvect_1 :::"RLSStruct"::: ) "," ($#l1_metric_1 :::"MetrStruct"::: ) ; aggr :::"RLSMetrStruct":::(# :::"carrier":::, :::"distance":::, :::"ZeroF":::, :::"addF":::, :::"Mult"::: #) -> ($#l1_vectmetr :::"RLSMetrStruct"::: ) ; end; registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v4_vectmetr :::"strict"::: ) for ($#l1_vectmetr :::"RLSMetrStruct"::: ) ; end; registrationlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "F" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set (Const "X")) "," (Set (Const "X")) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set ($#k1_numbers :::"REAL"::: ) ); let "O" be ($#m1_subset_1 :::"Element"::: ) "of" (Set (Const "X")); let "B" be ($#m1_subset_1 :::"BinOp":::) "of" (Set (Const "X")); let "G" be ($#m1_subset_1 :::"Function":::) "of" (Set ($#k2_zfmisc_1 :::"[:"::: ) (Set ($#k1_numbers :::"REAL"::: ) ) "," (Set (Const "X")) ($#k2_zfmisc_1 :::":]"::: ) ) "," (Set (Const "X")); cluster (Set ($#g1_vectmetr :::"RLSMetrStruct"::: ) "(#" "X" "," "F" "," "O" "," "B" "," "G" "#)" ) -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ; end; definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectmetr :::"RLSMetrStruct"::: ) ; attr "V" is :::"homogeneous"::: means :: VECTMETR:def 5 (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" "V" "holds" (Bool (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")) ")" ) "," (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "w")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k18_complex1 :::"abs"::: ) (Set (Var "r")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ")" ))))); end; :: deftheorem defines :::"homogeneous"::: VECTMETR:def 5 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectmetr :::"RLSMetrStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "V")) "is" ($#v5_vectmetr :::"homogeneous"::: ) ) "iff" (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")) ")" ) "," (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "w")) ")" ) ")" ) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k18_complex1 :::"abs"::: ) (Set (Var "r")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ")" ))))) ")" )); definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectmetr :::"RLSMetrStruct"::: ) ; attr "V" is :::"translatible"::: means :: VECTMETR:def 6 (Bool "for" (Set (Var "u")) "," (Set (Var "w")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" "V" "holds" (Bool (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set "(" (Set (Var "v")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "u")) ")" ) "," (Set "(" (Set (Var "w")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "u")) ")" ) ")" ))); end; :: deftheorem defines :::"translatible"::: VECTMETR:def 6 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectmetr :::"RLSMetrStruct"::: ) "holds" (Bool "(" (Bool (Set (Var "V")) "is" ($#v6_vectmetr :::"translatible"::: ) ) "iff" (Bool "for" (Set (Var "u")) "," (Set (Var "w")) "," (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set "(" (Set (Var "v")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "u")) ")" ) "," (Set "(" (Set (Var "w")) ($#k1_algstr_0 :::"+"::: ) (Set (Var "u")) ")" ) ")" ))) ")" )); definitionlet "V" be ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectmetr :::"RLSMetrStruct"::: ) ; let "v" be ($#m1_subset_1 :::"Element":::) "of" (Set (Const "V")); func :::"Norm"::: "v" -> ($#m1_subset_1 :::"Real":::) equals :: VECTMETR:def 7 (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set "(" ($#k4_struct_0 :::"0."::: ) "V" ")" ) "," "v" ")" ); end; :: deftheorem defines :::"Norm"::: VECTMETR:def 7 : (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#l1_vectmetr :::"RLSMetrStruct"::: ) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k3_vectmetr :::"Norm"::: ) (Set (Var "v"))) ($#r1_hidden :::"="::: ) (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) "," (Set (Var "v")) ")" )))); registration cluster ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v6_metric_1 :::"Reflexive"::: ) ($#v7_metric_1 :::"discerning"::: ) ($#v8_metric_1 :::"symmetric"::: ) ($#v9_metric_1 :::"triangle"::: ) ($#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"::: ) ($#v4_vectmetr :::"strict"::: ) ($#v5_vectmetr :::"homogeneous"::: ) ($#v6_vectmetr :::"translatible"::: ) for ($#l1_vectmetr :::"RLSMetrStruct"::: ) ; end; definitionmode RealLinearMetrSpace is ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v6_metric_1 :::"Reflexive"::: ) ($#v7_metric_1 :::"discerning"::: ) ($#v8_metric_1 :::"symmetric"::: ) ($#v9_metric_1 :::"triangle"::: ) ($#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"::: ) ($#v5_vectmetr :::"homogeneous"::: ) ($#v6_vectmetr :::"translatible"::: ) ($#l1_vectmetr :::"RLSMetrStruct"::: ) ; end; theorem :: VECTMETR:6 (Bool "for" (Set (Var "V")) "being" ($#~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"::: ) ($#v8_rlvect_1 :::"scalar-unital"::: ) ($#v5_vectmetr :::"homogeneous"::: ) ($#l1_vectmetr :::"RLSMetrStruct"::: ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) (Bool "for" (Set (Var "v")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k3_vectmetr :::"Norm"::: ) (Set "(" (Set (Var "r")) ($#k1_rlvect_1 :::"*"::: ) (Set (Var "v")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k18_complex1 :::"abs"::: ) (Set (Var "r")) ")" ) ($#k8_real_1 :::"*"::: ) (Set "(" ($#k3_vectmetr :::"Norm"::: ) (Set (Var "v")) ")" )))))) ; theorem :: VECTMETR:7 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v9_metric_1 :::"triangle"::: ) ($#v2_rlvect_1 :::"Abelian"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v6_vectmetr :::"translatible"::: ) ($#l1_vectmetr :::"RLSMetrStruct"::: ) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k3_vectmetr :::"Norm"::: ) (Set "(" (Set (Var "v")) ($#k3_rlvect_1 :::"+"::: ) (Set (Var "w")) ")" )) ($#r1_xxreal_0 :::"<="::: ) (Set (Set "(" ($#k3_vectmetr :::"Norm"::: ) (Set (Var "v")) ")" ) ($#k7_real_1 :::"+"::: ) (Set "(" ($#k3_vectmetr :::"Norm"::: ) (Set (Var "w")) ")" ))))) ; theorem :: VECTMETR:8 (Bool "for" (Set (Var "V")) "being" ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v13_algstr_0 :::"right_complementable"::: ) ($#v3_rlvect_1 :::"add-associative"::: ) ($#v4_rlvect_1 :::"right_zeroed"::: ) ($#v6_vectmetr :::"translatible"::: ) ($#l1_vectmetr :::"RLSMetrStruct"::: ) (Bool "for" (Set (Var "v")) "," (Set (Var "w")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set (Var "V")) "holds" (Bool (Set ($#k2_metric_1 :::"dist"::: ) "(" (Set (Var "v")) "," (Set (Var "w")) ")" ) ($#r1_hidden :::"="::: ) (Set ($#k3_vectmetr :::"Norm"::: ) (Set "(" (Set (Var "w")) ($#k5_algstr_0 :::"-"::: ) (Set (Var "v")) ")" ))))) ; definitionlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"RLMSpace"::: "n" -> ($#v4_vectmetr :::"strict"::: ) ($#l1_vectmetr :::"RealLinearMetrSpace":::) means :: VECTMETR:def 8 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k1_euclid :::"REAL"::: ) "n")) & (Bool (Set "the" ($#u1_metric_1 :::"distance"::: ) "of" it) ($#r1_funct_2 :::"="::: ) (Set ($#k13_euclid :::"Pitag_dist"::: ) "n")) & (Bool (Set ($#k4_struct_0 :::"0."::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) "n")) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "n") "holds" (Bool (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" it) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k7_euclid :::"+"::: ) (Set (Var "y")))) ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) "n") (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "the" ($#u1_rlvect_1 :::"Mult"::: ) "of" it) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "r")) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k9_euclid :::"*"::: ) (Set (Var "x"))))) ")" ) ")" ); end; :: deftheorem defines :::"RLMSpace"::: VECTMETR:def 8 : (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b2")) "being" ($#v4_vectmetr :::"strict"::: ) ($#l1_vectmetr :::"RealLinearMetrSpace":::) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k4_vectmetr :::"RLMSpace"::: ) (Set (Var "n")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n")))) & (Bool (Set "the" ($#u1_metric_1 :::"distance"::: ) "of" (Set (Var "b2"))) ($#r1_funct_2 :::"="::: ) (Set ($#k13_euclid :::"Pitag_dist"::: ) (Set (Var "n")))) & (Bool (Set ($#k4_struct_0 :::"0."::: ) (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k5_euclid :::"0*"::: ) (Set (Var "n")))) & (Bool "(" "for" (Set (Var "x")) "," (Set (Var "y")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) "holds" (Bool (Set (Set "the" ($#u1_algstr_0 :::"addF"::: ) "of" (Set (Var "b2"))) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "x")) "," (Set (Var "y")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "x")) ($#k7_euclid :::"+"::: ) (Set (Var "y")))) ")" ) & (Bool "(" "for" (Set (Var "x")) "being" ($#m2_finseq_2 :::"Element"::: ) "of" (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n"))) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k1_numbers :::"REAL"::: ) ) "holds" (Bool (Set (Set "the" ($#u1_rlvect_1 :::"Mult"::: ) "of" (Set (Var "b2"))) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "r")) "," (Set (Var "x")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k9_euclid :::"*"::: ) (Set (Var "x"))))) ")" ) ")" ) ")" ))); theorem :: VECTMETR:9 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#v2_funct_2 :::"onto"::: ) ($#v3_vectmetr :::"isometric"::: ) ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) (Set (Var "n")) ")" ) "," (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) (Set (Var "n")) ")" ) "holds" (Bool (Set ($#k10_xtuple_0 :::"rng"::: ) (Set (Var "f"))) ($#r1_hidden :::"="::: ) (Set ($#k1_euclid :::"REAL"::: ) (Set (Var "n")))))) ; begin definitionlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); func :::"IsomGroup"::: "n" -> ($#v15_algstr_0 :::"strict"::: ) ($#l3_algstr_0 :::"multMagma"::: ) means :: VECTMETR:def 9 (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" it) ($#r1_hidden :::"="::: ) (Set ($#k2_vectmetr :::"ISOM"::: ) (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) "n" ")" ))) & (Bool "(" "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k2_vectmetr :::"ISOM"::: ) (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) "n" ")" ))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k2_vectmetr :::"ISOM"::: ) (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) "n" ")" )))) "holds" (Bool (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" it) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set (Var "g")))) ")" ) ")" ); end; :: deftheorem defines :::"IsomGroup"::: VECTMETR:def 9 : (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "b2")) "being" ($#v15_algstr_0 :::"strict"::: ) ($#l3_algstr_0 :::"multMagma"::: ) "holds" (Bool "(" (Bool (Set (Var "b2")) ($#r1_hidden :::"="::: ) (Set ($#k5_vectmetr :::"IsomGroup"::: ) (Set (Var "n")))) "iff" (Bool "(" (Bool (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "b2"))) ($#r1_hidden :::"="::: ) (Set ($#k2_vectmetr :::"ISOM"::: ) (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) (Set (Var "n")) ")" ))) & (Bool "(" "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_hidden :::"Function":::) "st" (Bool (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set ($#k2_vectmetr :::"ISOM"::: ) (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) (Set (Var "n")) ")" ))) & (Bool (Set (Var "g")) ($#r2_hidden :::"in"::: ) (Set ($#k2_vectmetr :::"ISOM"::: ) (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) (Set (Var "n")) ")" )))) "holds" (Bool (Set (Set "the" ($#u2_algstr_0 :::"multF"::: ) "of" (Set (Var "b2"))) ($#k1_binop_1 :::"."::: ) "(" (Set (Var "f")) "," (Set (Var "g")) ")" ) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k3_relat_1 :::"*"::: ) (Set (Var "g")))) ")" ) ")" ) ")" ))); registrationlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k5_vectmetr :::"IsomGroup"::: ) "n") -> ($#~v2_struct_0 "non" ($#v2_struct_0 :::"empty"::: ) ) ($#v15_algstr_0 :::"strict"::: ) ; end; registrationlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); cluster (Set ($#k5_vectmetr :::"IsomGroup"::: ) "n") -> ($#v15_algstr_0 :::"strict"::: ) ($#v2_group_1 :::"Group-like"::: ) ($#v3_group_1 :::"associative"::: ) ; end; theorem :: VECTMETR:10 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) "holds" (Bool (Set ($#k1_group_1 :::"1_"::: ) (Set "(" ($#k5_vectmetr :::"IsomGroup"::: ) (Set (Var "n")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k3_struct_0 :::"id"::: ) (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) (Set (Var "n")) ")" )))) ; theorem :: VECTMETR:11 (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k5_vectmetr :::"IsomGroup"::: ) (Set (Var "n")) ")" ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"Function":::) "of" (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) (Set (Var "n")) ")" ) "," (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) (Set (Var "n")) ")" ) "st" (Bool (Bool (Set (Var "f")) ($#r1_hidden :::"="::: ) (Set (Var "g")))) "holds" (Bool (Set (Set (Var "f")) ($#k2_group_1 :::"""::: ) ) ($#r1_hidden :::"="::: ) (Set (Set (Var "g")) ($#k2_tops_2 :::"""::: ) ))))) ; definitionlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "G" be ($#m1_group_2 :::"Subgroup"::: ) "of" (Set ($#k5_vectmetr :::"IsomGroup"::: ) (Set (Const "n"))); func :::"SubIsomGroupRel"::: "G" -> ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) "n" ")" )) means :: VECTMETR:def 10 (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) "n" ")" ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "A")) "," (Set (Var "B")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) it) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" "G")) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "B"))) ")" )) ")" )); end; :: deftheorem defines :::"SubIsomGroupRel"::: VECTMETR:def 10 : (Bool "for" (Set (Var "n")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ) (Bool "for" (Set (Var "G")) "being" ($#m1_group_2 :::"Subgroup"::: ) "of" (Set ($#k5_vectmetr :::"IsomGroup"::: ) (Set (Var "n"))) (Bool "for" (Set (Var "b3")) "being" ($#m1_subset_1 :::"Relation":::) "of" (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) (Set (Var "n")) ")" )) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set ($#k6_vectmetr :::"SubIsomGroupRel"::: ) (Set (Var "G")))) "iff" (Bool "for" (Set (Var "A")) "," (Set (Var "B")) "being" ($#m1_subset_1 :::"Element":::) "of" (Set "(" ($#k4_vectmetr :::"RLMSpace"::: ) (Set (Var "n")) ")" ) "holds" (Bool "(" (Bool (Set ($#k1_domain_1 :::"["::: ) (Set (Var "A")) "," (Set (Var "B")) ($#k1_domain_1 :::"]"::: ) ) ($#r2_hidden :::"in"::: ) (Set (Var "b3"))) "iff" (Bool "ex" (Set (Var "f")) "being" ($#m1_hidden :::"Function":::) "st" (Bool "(" (Bool (Set (Var "f")) ($#r2_hidden :::"in"::: ) (Set "the" ($#u1_struct_0 :::"carrier"::: ) "of" (Set (Var "G")))) & (Bool (Set (Set (Var "f")) ($#k1_funct_1 :::"."::: ) (Set (Var "A"))) ($#r1_hidden :::"="::: ) (Set (Var "B"))) ")" )) ")" )) ")" )))); registrationlet "n" be ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k5_numbers :::"NAT"::: ) ); let "G" be ($#m1_group_2 :::"Subgroup"::: ) "of" (Set ($#k5_vectmetr :::"IsomGroup"::: ) (Set (Const "n"))); cluster (Set ($#k6_vectmetr :::"SubIsomGroupRel"::: ) "G") -> ($#v1_partfun1 :::"total"::: ) ($#v3_relat_2 :::"symmetric"::: ) ($#v8_relat_2 :::"transitive"::: ) ; end;