:: VFUNCT_2 semantic presentation begin definitionlet "M" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "V" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "f1" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); let "f2" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "M")) "," (Set (Const "V")); func "f1" :::"(#)"::: "f2" -> ($#m1_subset_1 :::"PartFunc":::) "of" "M" "," "V" means :: VFUNCT_2:def 1 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) "f1" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) "f2" ")" ))) & (Bool "(" "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "M" "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" "f1" ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set "(" "f2" ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"(#)"::: VFUNCT_2:def 1 : (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f2")) "," (Set (Var "b5")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b5"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ))) & (Bool "(" "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "M")) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b5"))))) "holds" (Bool (Set (Set (Var "b5")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))) ")" ) ")" ) ")" ))))); definitionlet "X" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "V" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "X")) "," (Set (Const "V")); let "z" be ($#m1_hidden :::"Complex":::); func "z" :::"(#)"::: "f" -> ($#m1_subset_1 :::"PartFunc":::) "of" "X" "," "V" means :: VFUNCT_2:def 2 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) "f")) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "X" "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) it))) "holds" (Bool (Set it ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set "z" ($#k1_clvect_1 :::"*"::: ) (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"(#)"::: VFUNCT_2:def 2 : (Bool "for" (Set (Var "X")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "X")) "," (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_hidden :::"Complex":::) (Bool "for" (Set (Var "b5")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "X")) "," (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "b5")) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b5"))) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "X")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b5"))))) "holds" (Bool (Set (Set (Var "b5")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ))) ")" ) ")" ) ")" )))))); theorem :: VFUNCT_2:1 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k1_tarski :::"}"::: ) ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" (Set (Var "f1")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" (Set (Var "f2")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ))))))) ; theorem :: VFUNCT_2:2 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k1_tarski :::"}"::: ) ))) & (Bool (Set (Set "(" ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k1_tarski :::"}"::: ) ))) ")" )))) ; theorem :: VFUNCT_2:3 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "z")) ($#r1_hidden :::"<>"::: ) (Set ($#k5_complex1 :::"0c"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set "(" ($#k4_struct_0 :::"0."::: ) (Set (Var "V")) ")" ) ($#k1_tarski :::"}"::: ) ))))))) ; theorem :: VFUNCT_2:4 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool (Set (Set (Var "f1")) ($#k1_vfunct_1 :::"+"::: ) (Set (Var "f2"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f2")) ($#k1_vfunct_1 :::"+"::: ) (Set (Var "f1"))))))) ; definitionlet "M" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "V" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "f1", "f2" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "M")) "," (Set (Const "V")); :: original: :::"+"::: redefine func "f1" :::"+"::: "f2" -> ($#m1_subset_1 :::"Element"::: ) "of" (Set bbbadK1_ZFMISC_1((Set bbbadK2_ZFMISC_1("M" "," (Set "the" ($#u1_struct_0 :::"U1"::: ) "of" "V"))))); commutativity (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "M")) "," (Set (Const "V")) "holds" (Bool (Set (Set (Var "f1")) ($#k1_vfunct_1 :::"+"::: ) (Set (Var "f2"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "f2")) ($#k1_vfunct_1 :::"+"::: ) (Set (Var "f1"))))) ; end; theorem :: VFUNCT_2:5 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f3"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f3")) ")" )))))) ; theorem :: VFUNCT_2:6 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f3"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set "(" (Set (Var "f2")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f3")) ")" ))))))) ; theorem :: VFUNCT_2:7 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f3"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f3")) ")" ) ($#k3_vfunct_2 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f3")) ")" ))))))) ; theorem :: VFUNCT_2:8 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "f3")) ($#k1_vfunct_2 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f3")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f1")) ")" ) ($#k3_vfunct_2 :::"+"::: ) (Set "(" (Set (Var "f3")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" ))))))) ; theorem :: VFUNCT_2:9 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f1")) ")" ) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2"))))))))) ; theorem :: VFUNCT_2:10 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set "(" (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" )))))))) ; theorem :: VFUNCT_2:11 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f3"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f3")) ")" ) ($#k2_vfunct_1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f3")) ")" ))))))) ; theorem :: VFUNCT_2:12 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f3")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f1")) ")" ) ($#k2_vfunct_1 :::"-"::: ) (Set "(" (Set (Var "f3")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f3")) ($#k1_vfunct_2 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" ))))))) ; theorem :: VFUNCT_2:13 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f1")) ")" ) ($#k3_vfunct_2 :::"+"::: ) (Set "(" (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" ))))))) ; theorem :: VFUNCT_2:14 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "z1")) "," (Set (Var "z2")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "z1")) ($#k9_complex1 :::"*"::: ) (Set (Var "z2")) ")" ) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "z1")) ($#k2_vfunct_2 :::"(#)"::: ) (Set "(" (Set (Var "z2")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f")) ")" ))))))) ; theorem :: VFUNCT_2:15 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f1")) ")" ) ($#k2_vfunct_1 :::"-"::: ) (Set "(" (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" ))))))) ; theorem :: VFUNCT_2:16 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k10_complex1 :::"-"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ($#k2_vfunct_2 :::"(#)"::: ) (Set "(" (Set (Var "f2")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f1")) ")" )))))) ; theorem :: VFUNCT_2:17 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f3")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f3"))))))) ; theorem :: VFUNCT_2:18 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool (Set (Set ($#k6_complex1 :::"1r"::: ) ) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f"))) ($#r2_relset_1 :::"="::: ) (Set (Var "f")))))) ; theorem :: VFUNCT_2:19 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f3")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f3"))))))) ; theorem :: VFUNCT_2:20 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool (Set (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f3")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f3"))))))) ; theorem :: VFUNCT_2:21 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set ($#k3_normsp_0 :::"||."::: ) (Set "(" (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k3_normsp_0 :::".||"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f1")) ($#k55_valued_1 :::".|"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f2")) ($#k3_normsp_0 :::".||"::: ) ))))))) ; theorem :: VFUNCT_2:22 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set ($#k3_normsp_0 :::"||."::: ) (Set "(" (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k3_normsp_0 :::".||"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k17_complex1 :::"|."::: ) (Set (Var "z")) ($#k17_complex1 :::".|"::: ) ) ($#k26_valued_1 :::"(#)"::: ) (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ))))))) ; theorem :: VFUNCT_2:23 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k10_complex1 :::"-"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f"))))))) ; theorem :: VFUNCT_2:24 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set "(" ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Var "f")))))) ; theorem :: VFUNCT_2:25 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set "(" ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" )))))) ; theorem :: VFUNCT_2:26 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set "(" ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2"))))))) ; theorem :: VFUNCT_2:27 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k3_vfunct_2 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2")))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) ")" ))))) ; theorem :: VFUNCT_2:28 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k1_vfunct_2 :::"(#)"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) ")" )))))) ; theorem :: VFUNCT_2:29 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set ($#k5_vfunct_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set ($#k3_normsp_0 :::"||."::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k3_normsp_0 :::".||"::: ) )) ")" ))))) ; theorem :: VFUNCT_2:30 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k2_vfunct_1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) ")" ))))) ; theorem :: VFUNCT_2:31 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set "(" (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" )))))))) ; theorem :: VFUNCT_2:32 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool "(" "(" (Bool (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v1_partfun1 :::"total"::: ) )) "implies" (Bool (Set (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2"))) "is" ($#v1_partfun1 :::"total"::: ) ) ")" & "(" (Bool (Bool (Set (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2"))) "is" ($#v1_partfun1 :::"total"::: ) )) "implies" (Bool "(" (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v1_partfun1 :::"total"::: ) ) ")" ) ")" & "(" (Bool (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v1_partfun1 :::"total"::: ) )) "implies" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) "is" ($#v1_partfun1 :::"total"::: ) ) ")" & "(" (Bool (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) "is" ($#v1_partfun1 :::"total"::: ) )) "implies" (Bool "(" (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v1_partfun1 :::"total"::: ) ) ")" ) ")" ")" )))) ; theorem :: VFUNCT_2:33 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v1_partfun1 :::"total"::: ) ) ")" ) "iff" (Bool (Set (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2"))) "is" ($#v1_partfun1 :::"total"::: ) ) ")" ))))) ; theorem :: VFUNCT_2:34 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) ) "iff" (Bool (Set (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f"))) "is" ($#v1_partfun1 :::"total"::: ) ) ")" ))))) ; theorem :: VFUNCT_2:35 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) ) "iff" (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f"))) "is" ($#v1_partfun1 :::"total"::: ) ) ")" )))) ; theorem :: VFUNCT_2:36 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) ) "iff" (Bool (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) "is" ($#v1_partfun1 :::"total"::: ) ) ")" )))) ; theorem :: VFUNCT_2:37 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "M")) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v1_partfun1 :::"total"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ) ($#k3_rlvect_1 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ) ($#k5_algstr_0 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ))) ")" ))))) ; theorem :: VFUNCT_2:38 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "M")) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v1_partfun1 :::"total"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" )))))))) ; theorem :: VFUNCT_2:39 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "M")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "z")) ($#k1_clvect_1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" )))))))) ; theorem :: VFUNCT_2:40 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "M")) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k4_algstr_0 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ))) & (Bool (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k1_funct_1 :::"."::: ) (Set (Var "x"))) ($#r1_hidden :::"="::: ) (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) )) ")" ))))) ; definitionlet "M" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "V" be ($#l2_clvect_1 :::"ComplexNormSpace":::); let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "M")) "," (Set (Const "V")); let "Y" be ($#m1_hidden :::"set"::: ) ; pred "f" :::"is_bounded_on"::: "Y" means :: VFUNCT_2:def 3 (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "M" "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set "Y" ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" )))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))))); end; :: deftheorem defines :::"is_bounded_on"::: VFUNCT_2:def 3 : (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool "(" (Bool (Set (Var "f")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y"))) "iff" (Bool "ex" (Set (Var "r")) "being" ($#m1_subset_1 :::"Real":::) "st" (Bool "for" (Set (Var "x")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "M")) "st" (Bool (Bool (Set (Var "x")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "Y")) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k1_normsp_0 :::"||."::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "x")) ")" ) ($#k1_normsp_0 :::".||"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "r"))))) ")" ))))); theorem :: VFUNCT_2:41 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "Y")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Var "f")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "X")))) "holds" (Bool (Set (Var "f")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y"))))))) ; theorem :: VFUNCT_2:42 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "X")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Var "f")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "X"))))))) ; theorem :: VFUNCT_2:43 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "holds" (Bool (Set (Set ($#k5_complex1 :::"0c"::: ) ) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f"))) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y"))))))) ; theorem :: VFUNCT_2:44 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f"))) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y")))))))) ; theorem :: VFUNCT_2:45 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y")))) "holds" (Bool "(" (Bool (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f"))) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y"))) ")" ))))) ; theorem :: VFUNCT_2:46 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2"))) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")))))))) ; theorem :: VFUNCT_2:47 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Var "f2")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2"))) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y"))))))))) ; theorem :: VFUNCT_2:48 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f2")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")))))))) ; theorem :: VFUNCT_2:49 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "X"))) & (Bool (Set (Var "f")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y")))) "holds" (Bool (Set (Var "f")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Y")))))))) ; theorem :: VFUNCT_2:50 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v3_funct_1 :::"constant"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v3_funct_1 :::"constant"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v3_funct_1 :::"constant"::: ) ) ")" ))))) ; theorem :: VFUNCT_2:51 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v3_funct_1 :::"constant"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_vfunct_2 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v3_funct_1 :::"constant"::: ) )))))) ; theorem :: VFUNCT_2:52 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )))))) ; theorem :: VFUNCT_2:53 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool "(" (Bool (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) ) & (Bool (Set (Set "(" ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) ) ")" ))))) ; theorem :: VFUNCT_2:54 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool (Set (Var "f")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y"))))))) ; theorem :: VFUNCT_2:55 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool "(" (Bool "(" "for" (Set (Var "z")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "z")) ($#k2_vfunct_2 :::"(#)"::: ) (Set (Var "f"))) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y"))) ")" ) & (Bool (Set ($#k5_vfunct_1 :::"-"::: ) (Set (Var "f"))) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "Y"))) & (Bool (Set (Set ($#k3_normsp_0 :::"||."::: ) (Set (Var "f")) ($#k3_normsp_0 :::".||"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ))))) ; theorem :: VFUNCT_2:56 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool (Set (Set (Var "f1")) ($#k3_vfunct_2 :::"+"::: ) (Set (Var "f2"))) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")))))))) ; theorem :: VFUNCT_2:57 (Bool "for" (Set (Var "M")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "V")) "being" ($#l2_clvect_1 :::"ComplexNormSpace":::) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "M")) "," (Set (Var "V")) (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) "st" (Bool (Bool (Set (Var "f1")) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool "(" (Bool (Set (Set (Var "f1")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f2"))) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")))) & (Bool (Set (Set (Var "f2")) ($#k2_vfunct_1 :::"-"::: ) (Set (Var "f1"))) ($#r1_vfunct_2 :::"is_bounded_on"::: ) (Set (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")))) ")" ))))) ;