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 LMX00:
  for S, T, W be RealNormSpace,
  f be Point of R_NormSpace_of_BoundedLinearOperators(S,W),
  g be Point of R_NormSpace_of_BoundedLinearOperators(T,W),
  I be LinearOperator of S, T
  st I is one-to-one onto isometric & f = g*I
  holds ||.f.|| = ||.g.||
  proof
    let S, T, W be RealNormSpace,
    f be Point of R_NormSpace_of_BoundedLinearOperators(S,W),
    g be Point of R_NormSpace_of_BoundedLinearOperators(T,W),
    I be LinearOperator of S, T;
    assume
    AS: I is one-to-one onto isometric & f = g*I;
    P1: dom I = the carrier of S by FUNCT_2:def 1;
    consider J be LinearOperator of T, S such that
    P2: J = I" & J is one-to-one onto isometric by AS,LM020;
    reconsider f0 = f as Lipschitzian LinearOperator of S, W
    by LOPBAN_1:def 9;
    reconsider g0 = g as Lipschitzian LinearOperator of T, W
    by LOPBAN_1:def 9;
    reconsider gI = g*I as Lipschitzian LinearOperator of S, W
    by LOPBAN_1:def 9,AS;
    Y1: for x be object holds
    x in {||.g0.t.|| where t is VECTOR of T : ||.t.|| <= 1 } iff
    x in {||.gI.w.|| where w is VECTOR of S : ||.w.|| <= 1 }
    proof
      let x be object;
      hereby
        assume x in {||.g0.t.|| where t is VECTOR of T : ||.t.|| <= 1 };
        then consider t be VECTOR of T such that
        B1: x = ||.g0.t.|| & ||.t.|| <= 1;
        set s = J.t;
        B2: gI.s = g0.(I.(J.t)) by P1,FUNCT_1:13
        .= g0.t by FUNCT_1:35,AS,P2;
        ||.s.|| <= 1 by B1,P2,LMMAZU;
        hence
        x in {||.gI.w.|| where w is VECTOR of S : ||.w.|| <= 1 } by B1,B2;
      end;
      assume x in {||.gI.w.|| where w is VECTOR of S : ||.w.|| <= 1 };
      then
      consider w be VECTOR of S such that
      B1: x = ||.gI.w.|| & ||.w.|| <= 1;
      set t = I.w;
      B2: gI.w = g0.t by P1,FUNCT_1:13;
      ||.t.|| <= 1 by B1,AS,LMMAZU;
      hence x in {||.g0.t.|| where t is VECTOR of T : ||.t.|| <= 1 } by B1,B2;
    end;
    X0: PreNorms(f0) = PreNorms(g0) by AS,Y1,TARSKI:2;
    X1: PreNorms(modetrans(f,S,W)) = PreNorms(f0) by LOPBAN_1:29
    .= PreNorms(modetrans(g,T,W)) by X0,LOPBAN_1:29;
    thus ||. f .|| = upper_bound PreNorms(modetrans(f,S,W)) by LOPBAN_1:def 13
    .= ||.g.|| by X1,LOPBAN_1:def 13;
  end;
