
theorem Th22:
for X be set, S be RealNormSpace, f,g be PartFunc of S,RNS_Real st
 f is_continuous_on X & g = ||.f.|| holds g is_continuous_on X
proof
    let X be set, S be RealNormSpace, f,g be PartFunc of S,RNS_Real;
    assume that
A1:  f is_continuous_on X and
A2:  g = ||.f.||;

    ||. f|X .|| = g|X by A2,NFCONT_1:35;
    hence thesis by A1,A2,Th21,NORMSP_0:def 2;
end;
