
theorem Th8:
  for X be ComplexNormSpace for f be Element of
  BoundedLinearOperators(X,X) holds f*FuncUnit(X)= f & FuncUnit(X)*f=f
proof
  let X be ComplexNormSpace;
  let f be Element of BoundedLinearOperators(X,X);
  (id the carrier of X) is Lipschitzian LinearOperator of X,X by Th3;
  then (id the carrier of X) is Element of BoundedLinearOperators(X,X) by
CLOPBAN1:def 7;
  then
A1: modetrans( (id (the carrier of X)),X,X) = (id the carrier of X) by
CLOPBAN1:def 9;
  hence f*FuncUnit(X) =modetrans(f,X,X) by Th6
    .=f by CLOPBAN1:def 9;
  thus FuncUnit(X)*f =modetrans(f,X,X) by A1,Th6
    .=f by CLOPBAN1:def 9;
end;
