reserve n,m for Nat;
reserve x,X,X1 for set;
reserve s,g,r,p for Real;
reserve S,T for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve s1,s2 for sequence of S;
reserve x0,x1,x2 for Point of S;
reserve Y for Subset of S;

theorem Th35:
  (||.f.||)|X = ||.(f|X).||
proof
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 S;
    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;
