let C be non empty set ; :: thesis: for c being Element of C
for f1, f2 being PartFunc of C,COMPLEX st f1 is total & f2 ^ is total holds
(f1 / f2) /. c = (f1 /. c) * ((f2 /. c) " )
let c be Element of C; :: thesis: for f1, f2 being PartFunc of C,COMPLEX st f1 is total & f2 ^ is total holds
(f1 / f2) /. c = (f1 /. c) * ((f2 /. c) " )
let f1, f2 be PartFunc of C,COMPLEX ; :: thesis: ( f1 is total & f2 ^ is total implies (f1 / f2) /. c = (f1 /. c) * ((f2 /. c) " ) )
assume ( f1 is total & f2 ^ is total ) ; :: thesis: (f1 / f2) /. c = (f1 /. c) * ((f2 /. c) " )
then ( f1 is total & f2 " {0c } = {} & f2 is total ) by Th74;
then f1 / f2 is total by Th75;
then dom (f1 / f2) = C by PARTFUN1:def 4;
hence (f1 / f2) /. c = (f1 /. c) * ((f2 /. c) " ) by Def1; :: thesis: verum