
theorem Th6:
  for X be ComplexNormSpace for f be Lipschitzian LinearOperator of X,X
  holds f*(id the carrier of X) = f & (id the carrier of X )*f=f
proof
  let X be ComplexNormSpace;
  reconsider ii=(id the carrier of X) as Lipschitzian LinearOperator of X,X
  by Th3;
  let f be Lipschitzian LinearOperator of X,X;
A1: now
    let x be VECTOR of X;
    thus ( (id the carrier of X)*f).x =( ii*f).x .=ii.(f.x) by Th4
      .=f.x;
  end;
  now
    let x be VECTOR of X;
    thus ( f*(id the carrier of X)).x =( f*ii).x .=f.(ii.x) by Th4
      .=f.x;
  end;
  hence thesis by A1,FUNCT_2:63;
end;
