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
  for X,Y be RealNormSpace for f be Element of BoundedLinearOperators
  (X,Y) holds f * FuncUnit(X) = f & FuncUnit(Y) * f = f
  proof
    let X,Y be RealNormSpace;
    let f be Element of BoundedLinearOperators(X,Y);
    (id the carrier of X) is Lipschitzian LinearOperator of X,X
      by LOPBAN_2:3; then
    (id the carrier of X) is Element of BoundedLinearOperators(X,X)
      by LOPBAN_1:def 9; then
    modetrans((id(the carrier of X)),X,X)
      = (id the carrier of X) by LOPBAN_1:def 11;
    hence f * FuncUnit(X) = modetrans(f,X,Y) by LPB2Th6
       .= f by LOPBAN_1:def 11;
    (id the carrier of Y) is Lipschitzian LinearOperator of Y,Y
      by LOPBAN_2:3; then
    (id the carrier of Y) is Element of BoundedLinearOperators(Y,Y)
      by LOPBAN_1:def 9; then
    modetrans( (id (the carrier of Y)),Y,Y)
      = (id the carrier of Y) by LOPBAN_1:def 11;
    hence FuncUnit(Y) * f = modetrans(f,X,Y) by LPB2Th6
       .= f by LOPBAN_1:def 11;
  end;
