let C be non empty set ; :: thesis: for c being Element of C
for V being RealNormSpace
for f2 being PartFunc of C,V
for f1 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 V being RealNormSpace
for f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL st f1 is total & f2 is total holds
(f1 (#) f2) /. c = (f1 . c) * (f2 /. c)
let V be RealNormSpace; :: thesis: for f2 being PartFunc of C,V
for f1 being PartFunc of C,REAL st f1 is total & f2 is total holds
(f1 (#) f2) /. c = (f1 . c) * (f2 /. c)
let f2 be PartFunc of C,V; :: thesis: for f1 being PartFunc of C,REAL st f1 is total & f2 is total holds
(f1 (#) f2) /. c = (f1 . c) * (f2 /. c)
let f1 be PartFunc of C,REAL; :: 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 dom (f1 (#) f2) = C by PARTFUN1:def 2;
hence (f1 (#) f2) /. c = (f1 . c) * (f2 /. c) by Def3; :: thesis: verum