let C be non empty set ; :: thesis: for c being Element of C

for f1, f2 being PartFunc of C,REAL 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,REAL st f1 is total & f2 ^ is total holds

(f1 / f2) . c = (f1 . c) * ((f2 . c) ")

let f1, f2 be PartFunc of C,REAL; :: thesis: ( f1 is total & f2 ^ is total implies (f1 / f2) . c = (f1 . c) * ((f2 . c) ") )

assume that

A1: f1 is total and

A2: f2 ^ is total ; :: thesis: (f1 / f2) . c = (f1 . c) * ((f2 . c) ")

A3: f2 is total by A2, Th54;

f2 " {0} = {} by A2, Th54;

then f1 / f2 is total by A1, A3, Th55;

then dom (f1 / f2) = C ;

hence (f1 / f2) . c = (f1 . c) * ((f2 . c) ") by Def1; :: thesis: verum

for f1, f2 being PartFunc of C,REAL 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,REAL st f1 is total & f2 ^ is total holds

(f1 / f2) . c = (f1 . c) * ((f2 . c) ")

let f1, f2 be PartFunc of C,REAL; :: thesis: ( f1 is total & f2 ^ is total implies (f1 / f2) . c = (f1 . c) * ((f2 . c) ") )

assume that

A1: f1 is total and

A2: f2 ^ is total ; :: thesis: (f1 / f2) . c = (f1 . c) * ((f2 . c) ")

A3: f2 is total by A2, Th54;

f2 " {0} = {} by A2, Th54;

then f1 / f2 is total by A1, A3, Th55;

then dom (f1 / f2) = C ;

hence (f1 / f2) . c = (f1 . c) * ((f2 . c) ") by Def1; :: thesis: verum