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 LPB2Th6:
  for X,Y be RealNormSpace for f be Lipschitzian LinearOperator of X,Y
  holds f * (id the carrier of X) = f & (id the carrier of Y) * f = f
  proof
    let X,Y be RealNormSpace;
    reconsider ii = id the carrier of X
      as Lipschitzian LinearOperator of X,X by LOPBAN_2:3;
    reconsider jj = id the carrier of Y
      as Lipschitzian LinearOperator of Y,Y by LOPBAN_2:3;
    let f be Lipschitzian LinearOperator of X,Y;
    A1: now
      let x be VECTOR of X;
      thus ((id the carrier of Y)*f).x = jj.(f.x) by FUNCT_2:15
        .= f.x;
    end;
    now
      let x be VECTOR of X;
      thus (f*(id the carrier of X)).x = f.(ii.x) by FUNCT_2:15
        .= f.x;
    end;
    hence thesis by A1,FUNCT_2:63;
  end;
