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) & (f1 - f2) . c = (f1 . c) - (f2 . c) & (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) & (f1 - f2) . c = (f1 . c) - (f2 . c) & (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) & (f1 - f2) . c = (f1 . c) - (f2 . c) & (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) & (f1 - f2) . c = (f1 . c) - (f2 . c) & (f1 (#) f2) . c = (f1 . c) * (f2 . c) )
f1 + f2 is total by A1, A2;
then dom (f1 + f2) = C ;
hence (f1 + f2) . c = (f1 . c) + (f2 . c) by VALUED_1:def 1; :: thesis: ( (f1 - f2) . c = (f1 . c) - (f2 . c) & (f1 (#) f2) . c = (f1 . c) * (f2 . c) )
f1 - f2 is total by A1, A2;
then dom (f1 - f2) = C ;
hence (f1 - f2) . c = (f1 . c) - (f2 . c) by VALUED_1:13; :: thesis: (f1 (#) f2) . c = (f1 . c) * (f2 . c)
thus (f1 (#) f2) . c = (f1 . c) * (f2 . c) by VALUED_1:5; :: thesis: verum