let M be non empty set ; :: thesis: for V being ComplexNormSpace
for f being PartFunc of M, the carrier of V holds
( f is total iff ||.f.|| is total )
let V be ComplexNormSpace; :: thesis: for f being PartFunc of M, the carrier of V holds
( f is total iff ||.f.|| is total )
let f be PartFunc of M, the carrier of V; :: thesis: ( f is total iff ||.f.|| is total )
thus ( f is total implies ||.f.|| is total ) :: thesis: ( ||.f.|| is total implies f is total )

proof

assume f is total ; :: thesis: ||.f.|| is total
then dom f = M by PARTFUN1:def 4;
hence dom ||.f.|| = M by NORMSP_0:def 3; :: according to PARTFUN1:def 4 :: thesis: verum

end;

assume ||.f.|| is total ; :: thesis: f is total
then dom ||.f.|| = M by PARTFUN1:def 4;
hence dom f = M by NORMSP_0:def 3; :: according to PARTFUN1:def 4 :: thesis: verum