:: CFUNCT_1 semantic presentation begin definitionmode Complex is ($#v1_xcmplx_0 :::"complex"::: ) ($#m1_hidden :::"number"::: ) ; end; definitionlet "C" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f1", "f2" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); func "f1" :::"/"::: "f2" -> ($#m1_subset_1 :::"PartFunc":::) "of" "C" "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) means :: CFUNCT_1:def 1 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) "f1" ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set "(" ($#k1_relset_1 :::"dom"::: ) "f2" ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" "f2" ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ))) & (Bool "(" "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "C" "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" ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k9_complex1 :::"*"::: ) (Set "(" (Set "(" "f2" ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k12_complex1 :::"""::: ) ")" ))) ")" ) ")" ); end; :: deftheorem defines :::"/"::: CFUNCT_1:def 1 : (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "b4")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b4")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f2")))) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b4"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" (Set (Var "f2")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ))) & (Bool "(" "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b4"))))) "holds" (Bool (Set (Set (Var "b4")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k9_complex1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k12_complex1 :::"""::: ) ")" ))) ")" ) ")" ) ")" ))); definitionlet "C" be ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) ; let "f" be ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Const "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ); func "f" :::"^"::: -> ($#m1_subset_1 :::"PartFunc":::) "of" "C" "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) means :: CFUNCT_1:def 2 (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) it) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) "f" ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" "f" ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" "C" "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 "(" "f" ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k12_complex1 :::"""::: ) )) ")" ) ")" ); end; :: deftheorem defines :::"^"::: CFUNCT_1:def 2 : (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "b3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "b3")) ($#r1_hidden :::"="::: ) (Set (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) )) "iff" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ))) & (Bool "(" "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "b3"))))) "holds" (Bool (Set (Set (Var "b3")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k12_complex1 :::"""::: ) )) ")" ) ")" ) ")" ))); theorem :: CFUNCT_1:1 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" )) ($#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 "C")) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k8_complex1 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))) ")" ) ")" ))) ; theorem :: CFUNCT_1:2 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f1")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f2")) ")" ))) & (Bool "(" "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k11_complex1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))) ")" ) ")" ))) ; theorem :: CFUNCT_1:3 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" )) ($#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 "C")) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k9_complex1 :::"*"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))) ")" ) ")" ))) ; theorem :: CFUNCT_1:4 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k9_complex1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))) ")" ) ")" )))) ; theorem :: CFUNCT_1:5 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "f")) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")))) & (Bool "(" "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k10_complex1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))) ")" ) ")" ))) ; theorem :: CFUNCT_1:6 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "g")) ($#k2_cfunct_1 :::"^"::: ) ")" )) ($#r1_tarski :::"c="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")))) & (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")) ")" ) ($#k9_subset_1 :::"/\"::: ) (Set "(" (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" (Set (Var "g")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ")" )) ($#r1_hidden :::"="::: ) (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "g")) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" (Set (Var "g")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ))) ")" ))) ; theorem :: CFUNCT_1:7 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ")" ) ($#k7_subset_1 :::"\"::: ) (Set "(" (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#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 ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) ) ")" ) ")" ))))) ; theorem :: CFUNCT_1:8 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ")" )))) "holds" (Bool (Set (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))))) ; theorem :: CFUNCT_1:9 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )))) ; theorem :: CFUNCT_1:10 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f")) ($#k55_valued_1 :::".|"::: ) ) ($#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 ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) ))) & (Bool (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#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 ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) ))) ")" ))) ; theorem :: CFUNCT_1:11 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set "(" (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ($#k2_cfunct_1 :::"^"::: ) ")" )) ($#r1_hidden :::"="::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ")" ) ")" ))))) ; theorem :: CFUNCT_1:12 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#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 ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )))))) ; begin theorem :: CFUNCT_1:13 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set (Var "f3"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f3")) ")" ))))) ; theorem :: CFUNCT_1:14 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f3"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f2")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f3")) ")" ))))) ; theorem :: CFUNCT_1:15 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f3"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f3")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f3")) ")" ))))) ; theorem :: CFUNCT_1:16 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f3")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "f3")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f3")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f1")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "f3")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ))))) ; theorem :: CFUNCT_1:17 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f1")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2"))))))) ; theorem :: CFUNCT_1:18 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" )))))) ; theorem :: CFUNCT_1:19 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f3"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f3")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f3")) ")" ))))) ; theorem :: CFUNCT_1:20 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f3")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f3")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f1")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "f3")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f3")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ))))) ; theorem :: CFUNCT_1:21 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f1")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" )))))) ; theorem :: CFUNCT_1:22 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "," (Set (Var "q")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k9_complex1 :::"*"::: ) (Set (Var "q")) ")" ) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "q")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" )))))) ; theorem :: CFUNCT_1:23 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f1")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" )))))) ; theorem :: CFUNCT_1:24 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k10_complex1 :::"-"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f2")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f1")) ")" ))))) ; theorem :: CFUNCT_1:25 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f3")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set (Var "f3")))))) ; theorem :: CFUNCT_1:26 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set ($#k6_complex1 :::"1r"::: ) ) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f"))) ($#r2_relset_1 :::"="::: ) (Set (Var "f"))))) ; theorem :: CFUNCT_1:27 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f3")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set (Var "f3")))))) ; theorem :: CFUNCT_1:28 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "," (Set (Var "f3")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f3")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set (Var "f3")))))) ; theorem :: CFUNCT_1:29 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set ($#k55_valued_1 :::"|."::: ) (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k55_valued_1 :::".|"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f1")) ($#k55_valued_1 :::".|"::: ) ) ($#k20_valued_1 :::"(#)"::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f2")) ($#k55_valued_1 :::".|"::: ) ))))) ; theorem :: CFUNCT_1:30 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set ($#k55_valued_1 :::"|."::: ) (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k55_valued_1 :::".|"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k17_complex1 :::"|."::: ) (Set (Var "r")) ($#k17_complex1 :::".|"::: ) ) ($#k26_valued_1 :::"(#)"::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f")) ($#k55_valued_1 :::".|"::: ) )))))) ; theorem :: CFUNCT_1:31 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set ($#k31_valued_1 :::"-"::: ) (Set (Var "f"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k10_complex1 :::"-"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")))))) ; theorem :: CFUNCT_1:32 canceled; theorem :: CFUNCT_1:33 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")))))) ; theorem :: CFUNCT_1:34 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ($#k2_cfunct_1 :::"^"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ")" ))))) ; theorem :: CFUNCT_1:35 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_cfunct_1 :::"^"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f2")) ($#k2_cfunct_1 :::"^"::: ) ")" ))))) ; theorem :: CFUNCT_1:36 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "r")) ($#r1_hidden :::"<>"::: ) (Set ($#k6_numbers :::"0"::: ) ))) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_cfunct_1 :::"^"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k12_complex1 :::"""::: ) ")" ) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ")" )))))) ; theorem :: CFUNCT_1:37 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_cfunct_1 :::"^"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k10_complex1 :::"-"::: ) (Set ($#k6_complex1 :::"1r"::: ) ) ")" ) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ")" ))))) ; theorem :: CFUNCT_1:38 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f")) ($#k55_valued_1 :::".|"::: ) ) ($#k6_rfunct_1 :::"^"::: ) ) ($#r2_relset_1 :::"="::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set "(" (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ($#k55_valued_1 :::".|"::: ) )))) ; theorem :: CFUNCT_1:39 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "f")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "g"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "g")) ($#k2_cfunct_1 :::"^"::: ) ")" ))))) ; theorem :: CFUNCT_1:40 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "," (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "g")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f"))))))) ; theorem :: CFUNCT_1:41 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "g")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "g"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "g")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ")" ))))) ; theorem :: CFUNCT_1:42 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "f1")) "," (Set (Var "g1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "g")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "g1")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f1")) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set "(" (Set (Var "g")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "g1")) ")" ))))) ; theorem :: CFUNCT_1:43 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) ($#k2_cfunct_1 :::"^"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f1")))))) ; theorem :: CFUNCT_1:44 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "g")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "g")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f1")) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f2")))))) ; theorem :: CFUNCT_1:45 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "g")) "," (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set (Var "g")) ($#k1_cfunct_1 :::"/"::: ) (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f2")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "g")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "f2")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f1")))))) ; theorem :: CFUNCT_1:46 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set ($#k31_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "g")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "g")))) & (Bool (Set (Set (Var "f")) ($#k1_cfunct_1 :::"/"::: ) (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "g")) ")" )) ($#r2_relset_1 :::"="::: ) (Set ($#k31_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "g")) ")" ))) ")" ))) ; theorem :: CFUNCT_1:47 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f")))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f")))) ")" ))) ; theorem :: CFUNCT_1:48 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "," (Set (Var "g1")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "g1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "g")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "g1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set "(" (Set (Var "f")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ))))) ; theorem :: CFUNCT_1:49 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "," (Set (Var "g")) "," (Set (Var "f1")) "," (Set (Var "g1")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "g")) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "g1")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "g1")) ($#k2_partfun1 :::"|"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set "(" (Set (Var "g1")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ")" ) ")" ) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set "(" (Set (Var "g")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f1")) ")" ))))) ; theorem :: CFUNCT_1:50 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f")) "," (Set (Var "g1")) "," (Set (Var "g")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "g1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "g")) ")" )) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "g1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set "(" (Set (Var "f")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "g")) ")" ))))) ; theorem :: CFUNCT_1:51 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set ($#k55_valued_1 :::"|."::: ) (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) ($#k55_valued_1 :::".|"::: ) ) ($#r2_relset_1 :::"="::: ) (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f1")) ($#k55_valued_1 :::".|"::: ) ) ($#k3_rfunct_1 :::"/"::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f2")) ($#k55_valued_1 :::".|"::: ) ))))) ; theorem :: CFUNCT_1:52 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) ")" )))) ; theorem :: CFUNCT_1:53 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) ")" )))) ; theorem :: CFUNCT_1:54 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set ($#k31_valued_1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Set "(" (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k2_cfunct_1 :::"^"::: ) )) & (Bool (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f")) ($#k55_valued_1 :::".|"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set ($#k55_valued_1 :::"|."::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k55_valued_1 :::".|"::: ) )) ")" )))) ; theorem :: CFUNCT_1:55 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) ")" )))) ; theorem :: CFUNCT_1:56 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f2")))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set "(" (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))) ")" )))) ; theorem :: CFUNCT_1:57 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) ($#r2_relset_1 :::"="::: ) (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set "(" (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X")) ")" ))))))) ; begin theorem :: CFUNCT_1:58 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "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")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2"))) "is" ($#v1_partfun1 :::"total"::: ) ) ")" & "(" (Bool (Bool (Set (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2"))) "is" ($#v1_partfun1 :::"total"::: ) )) "implies" (Bool "(" (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v1_partfun1 :::"total"::: ) ) ")" ) ")" & "(" (Bool (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v1_partfun1 :::"total"::: ) )) "implies" (Bool (Set (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2"))) "is" ($#v1_partfun1 :::"total"::: ) ) ")" & "(" (Bool (Bool (Set (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2"))) "is" ($#v1_partfun1 :::"total"::: ) )) "implies" (Bool "(" (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v1_partfun1 :::"total"::: ) ) ")" ) ")" & "(" (Bool (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v1_partfun1 :::"total"::: ) )) "implies" (Bool (Set (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2"))) "is" ($#v1_partfun1 :::"total"::: ) ) ")" & "(" (Bool (Bool (Set (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2"))) "is" ($#v1_partfun1 :::"total"::: ) )) "implies" (Bool "(" (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Var "f2")) "is" ($#v1_partfun1 :::"total"::: ) ) ")" ) ")" ")" ))) ; theorem :: CFUNCT_1:59 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) ) "iff" (Bool (Set (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f"))) "is" ($#v1_partfun1 :::"total"::: ) ) ")" )))) ; theorem :: CFUNCT_1:60 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) ) "iff" (Bool (Set ($#k31_valued_1 :::"-"::: ) (Set (Var "f"))) "is" ($#v1_partfun1 :::"total"::: ) ) ")" ))) ; theorem :: CFUNCT_1:61 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) ) "iff" (Bool (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f")) ($#k55_valued_1 :::".|"::: ) ) "is" ($#v1_partfun1 :::"total"::: ) ) ")" ))) ; theorem :: CFUNCT_1:62 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ) "is" ($#v1_partfun1 :::"total"::: ) ) "iff" (Bool "(" (Bool (Set (Set (Var "f")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) ) ")" ) ")" ))) ; theorem :: CFUNCT_1:63 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool "(" (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k8_relset_1 :::"""::: ) (Set ($#k1_tarski :::"{"::: ) (Set ($#k6_numbers :::"0"::: ) ) ($#k1_tarski :::"}"::: ) )) ($#r1_hidden :::"="::: ) (Set ($#k1_xboole_0 :::"{}"::: ) )) & (Bool (Set (Var "f2")) "is" ($#v1_partfun1 :::"total"::: ) ) ")" ) "iff" (Bool (Set (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f2"))) "is" ($#v1_partfun1 :::"total"::: ) ) ")" ))) ; theorem :: CFUNCT_1:64 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "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")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k8_complex1 :::"+"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k11_complex1 :::"-"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k9_complex1 :::"*"::: ) (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))) ")" )))) ; theorem :: CFUNCT_1:65 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set (Var "r")) ($#k9_complex1 :::"*"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))))))) ; theorem :: CFUNCT_1:66 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f")) "is" ($#v1_partfun1 :::"total"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k10_complex1 :::"-"::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ))) & (Bool (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f")) ($#k55_valued_1 :::".|"::: ) ) ($#k1_funct_1 :::"."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k17_complex1 :::".|"::: ) )) ")" )))) ; theorem :: CFUNCT_1:67 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ) "is" ($#v1_partfun1 :::"total"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f")) ($#k2_cfunct_1 :::"^"::: ) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k12_complex1 :::"""::: ) ))))) ; theorem :: CFUNCT_1:68 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "f1")) "is" ($#v1_partfun1 :::"total"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_cfunct_1 :::"^"::: ) ) "is" ($#v1_partfun1 :::"total"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k1_cfunct_1 :::"/"::: ) (Set (Var "f2")) ")" ) ($#k7_partfun1 :::"/."::: ) (Set (Var "c"))) ($#r1_hidden :::"="::: ) (Set (Set "(" (Set (Var "f1")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k9_complex1 :::"*"::: ) (Set "(" (Set "(" (Set (Var "f2")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k12_complex1 :::"""::: ) ")" )))))) ; begin theorem :: CFUNCT_1:69 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) "iff" (Bool "ex" (Set (Var "p")) "being" ($#v1_xreal_0 :::"real"::: ) ($#m1_hidden :::"number"::: ) "st" (Bool "for" (Set (Var "c")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set (Var "C")) "st" (Bool (Bool (Set (Var "c")) ($#r2_hidden :::"in"::: ) (Set (Set (Var "Y")) ($#k9_subset_1 :::"/\"::: ) (Set "(" ($#k1_relset_1 :::"dom"::: ) (Set (Var "f")) ")" )))) "holds" (Bool (Set ($#k17_complex1 :::"|."::: ) (Set "(" (Set (Var "f")) ($#k7_partfun1 :::"/."::: ) (Set (Var "c")) ")" ) ($#k17_complex1 :::".|"::: ) ) ($#r1_xxreal_0 :::"<="::: ) (Set (Var "p"))))) ")" )))) ; theorem :: CFUNCT_1:70 (Bool "for" (Set (Var "Y")) "," (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "Y")) ($#r1_tarski :::"c="::: ) (Set (Var "X"))) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )))) ; theorem :: CFUNCT_1:71 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Var "X")) ($#r1_xboole_0 :::"misses"::: ) (Set ($#k1_relset_1 :::"dom"::: ) (Set (Var "f"))))) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )))) ; theorem :: CFUNCT_1:72 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ))))) ; theorem :: CFUNCT_1:73 (Bool "for" (Set (Var "X")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f")) ($#k55_valued_1 :::".|"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v2_seq_2 :::"bounded_below"::: ) )))) ; theorem :: CFUNCT_1:74 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f")) ($#k55_valued_1 :::".|"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" )))) ; theorem :: CFUNCT_1:75 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v1_comseq_2 :::"bounded"::: ) )))) ; theorem :: CFUNCT_1:76 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" )))) ; theorem :: CFUNCT_1:77 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k2_xboole_0 :::"\/"::: ) (Set (Var "Y")) ")" )) "is" ($#v1_comseq_2 :::"bounded"::: ) )))) ; theorem :: CFUNCT_1:78 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#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 "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v3_funct_1 :::"constant"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v3_funct_1 :::"constant"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v3_funct_1 :::"constant"::: ) ) ")" )))) ; theorem :: CFUNCT_1:79 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) (Bool "for" (Set (Var "q")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "q")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) ))))) ; theorem :: CFUNCT_1:80 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool "(" (Bool (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f")) ($#k55_valued_1 :::".|"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) ) & (Bool (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) ) ")" )))) ; theorem :: CFUNCT_1:81 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) )))) ; theorem :: CFUNCT_1:82 (Bool "for" (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool "(" (Bool "(" "for" (Set (Var "r")) "being" ($#m1_subset_1 :::"Element"::: ) "of" (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool (Set (Set "(" (Set (Var "r")) ($#k25_valued_1 :::"(#)"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ) & (Bool (Set (Set "(" ($#k31_valued_1 :::"-"::: ) (Set (Var "f")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f")) ($#k55_valued_1 :::".|"::: ) ) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" )))) ; theorem :: CFUNCT_1:83 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool (Set (Set "(" (Set (Var "f1")) ($#k2_valued_1 :::"+"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v1_comseq_2 :::"bounded"::: ) )))) ; theorem :: CFUNCT_1:84 (Bool "for" (Set (Var "X")) "," (Set (Var "Y")) "being" ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f1")) "," (Set (Var "f2")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "st" (Bool (Bool (Set (Set (Var "f1")) ($#k2_partfun1 :::"|"::: ) (Set (Var "X"))) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set (Var "f2")) ($#k2_partfun1 :::"|"::: ) (Set (Var "Y"))) "is" ($#v3_funct_1 :::"constant"::: ) )) "holds" (Bool "(" (Bool (Set (Set "(" (Set (Var "f1")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set (Var "f2")) ($#k46_valued_1 :::"-"::: ) (Set (Var "f1")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) & (Bool (Set (Set "(" (Set (Var "f1")) ($#k19_valued_1 :::"(#)"::: ) (Set (Var "f2")) ")" ) ($#k2_partfun1 :::"|"::: ) (Set "(" (Set (Var "X")) ($#k3_xboole_0 :::"/\"::: ) (Set (Var "Y")) ")" )) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" )))) ; theorem :: CFUNCT_1:85 (Bool "for" (Set (Var "C")) "being" ($#~v1_xboole_0 "non" ($#v1_xboole_0 :::"empty"::: ) ) ($#m1_hidden :::"set"::: ) (Bool "for" (Set (Var "f")) "being" ($#m1_subset_1 :::"PartFunc":::) "of" (Set (Var "C")) "," (Set ($#k2_numbers :::"COMPLEX"::: ) ) "holds" (Bool "(" (Bool (Set ($#k55_valued_1 :::"|."::: ) (Set (Var "f")) ($#k55_valued_1 :::".|"::: ) ) "is" ($#v1_comseq_2 :::"bounded"::: ) ) "iff" (Bool (Set (Var "f")) "is" ($#v1_comseq_2 :::"bounded"::: ) ) ")" ))) ;