reserve S,T,W,Y for RealNormSpace;
reserve f,f1,f2 for PartFunc of S,T;
reserve Z for Subset of S;
reserve i,n for Nat;

theorem LM010:
  for I be LinearOperator of S,T,
      x be Point of S st I is isometric holds
    I is_continuous_in x
  proof
    let I be LinearOperator of S, T,
        x0 be Point of S;
    assume AS1: I is isometric;
    P1: dom I = the carrier of S by FUNCT_2:def 1;
    for r be Real st 0<r ex s be Real st
    0 < s & for x1 be Point of S st x1 in dom I & ||. x1- x0 .|| < s
    holds ||. I/.x1-I/.x0 .|| < r
    proof
      let r be Real;
      assume P2: 0 < r;
      take s = r;
      thus 0 < s by P2;
      let x1 be Point of S;
      assume P3: x1 in dom I & ||. x1- x0 .|| < s;
      thus thesis by AS1,P3;
    end;
    hence thesis by NFCONT_1:7,P1;
  end;
