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 LPB2Th4:
  for X,Y,Z be RealNormSpace
  for g be Lipschitzian LinearOperator of X,Y,
      f be Lipschitzian LinearOperator of Y,Z,
      h be Lipschitzian LinearOperator of X,Z
  holds h = f*g iff for x be VECTOR of X holds h.x = f.(g.x)
  proof
    let X,Y,Z be RealNormSpace;
    let g be Lipschitzian LinearOperator of X,Y,
        f be Lipschitzian LinearOperator of Y,Z,
        h be Lipschitzian LinearOperator of X,Z;
    now
      assume
  A1: for x being VECTOR of X holds h.x = f.(g.x);
      now
        let x be VECTOR of X;
        thus (f*g).x = f.(g.x) by FUNCT_2:15
                    .= h.x by A1;
      end;
      hence h = f*g by FUNCT_2:63;
    end;
    hence thesis by FUNCT_2:15;
  end;
