
theorem Th4:
  for X be ComplexNormSpace for f,g,h be Lipschitzian LinearOperator of
  X,X holds h = f*g iff for x be VECTOR of X holds h.x = f.(g.x)
proof
  let X be ComplexNormSpace;
  let f,g,h be Lipschitzian LinearOperator of X,X;
  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;
