let C be non empty set ; :: thesis: for c being Element of C
for V being RealNormSpace
for f1, f2 being PartFunc of C,V st f1 is total & f2 is total holds
( (f1 + f2) /. c = (f1 /. c) + (f2 /. c) & (f1 - f2) /. c = (f1 /. c) - (f2 /. c) )
let c be Element of C; :: thesis: for V being RealNormSpace
for f1, f2 being PartFunc of C,V st f1 is total & f2 is total holds
( (f1 + f2) /. c = (f1 /. c) + (f2 /. c) & (f1 - f2) /. c = (f1 /. c) - (f2 /. c) )
let V be RealNormSpace; :: thesis: for f1, f2 being PartFunc of C,V st f1 is total & f2 is total holds
( (f1 + f2) /. c = (f1 /. c) + (f2 /. c) & (f1 - f2) /. c = (f1 /. c) - (f2 /. c) )
let f1, f2 be PartFunc of C,V; :: thesis: ( f1 is total & f2 is total implies ( (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) )
then dom (f1 + f2) = C by PARTFUN1:def 2;
hence (f1 + f2) /. c = (f1 /. c) + (f2 /. c) by Def1; :: thesis: (f1 - f2) /. c = (f1 /. c) - (f2 /. c)
dom (f1 - f2) = C by A1, PARTFUN1:def 2;
hence (f1 - f2) /. c = (f1 /. c) - (f2 /. c) by Def2; :: thesis: verum