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) & (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,COMPLEX 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,COMPLEX; :: 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 A1: ( f1 is total & 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) )
then f1 + f2 is total ;
then dom (f1 + f2) = C ;
hence (f1 + f2) /. c = (f1 /. c) + (f2 /. c) by Th1; :: thesis: ( (f1 - f2) /. c = (f1 /. c) - (f2 /. c) & (f1 (#) f2) /. c = (f1 /. c) * (f2 /. c) )
f1 - f2 is total by A1;
then dom (f1 - f2) = C ;
hence (f1 - f2) /. c = (f1 /. c) - (f2 /. c) by Th2; :: thesis: (f1 (#) f2) /. c = (f1 /. c) * (f2 /. c)
f1 (#) f2 is total by A1;
then dom (f1 (#) f2) = C ;
hence (f1 (#) f2) /. c = (f1 /. c) * (f2 /. c) by Th3; :: thesis: verum