reserve n,m for Element of NAT;
reserve r,s for Real;
reserve z for Complex;
reserve CNS,CNS1,CNS2 for ComplexNormSpace;
reserve RNS for RealNormSpace;
reserve X,X1 for set;

theorem Th90:
  for f be PartFunc of CNS1,CNS2 holds (||.f.||)|X = ||.(f|X).||
proof
  let f be PartFunc of CNS1,CNS2;
A1: dom ((||.f.||)|X) = dom (||.f.||) /\ X by RELAT_1:61
    .= dom f /\ X by NORMSP_0:def 3
    .= dom (f|X) by RELAT_1:61
    .= dom ( ||.(f|X).||) by NORMSP_0:def 3;
  now
    let c be Point of CNS1;
    assume
A2: c in dom ((||.f.||)|X);
    then
A3: c in dom (f|X) by A1,NORMSP_0:def 3;
    c in dom (||.f.||) /\ X by A2,RELAT_1:61;
    then
A4: c in dom (||.f.||) by XBOOLE_0:def 4;
    thus ((||.f.||)|X).c = (||.f.||).c by A2,FUNCT_1:47
      .= ||.f/.c.|| by A4,NORMSP_0:def 3
      .= ||.(f|X)/.c.|| by A3,PARTFUN2:15
      .= ||.(f|X).||.c by A1,A2,NORMSP_0:def 3;
  end;
  hence thesis by A1,PARTFUN1:5;
end;
