theorem
  for F being Function of I,the carrier' of C holds f*F = (I-->f)"*"F
proof
  let F be Function of I,the carrier' of C;
  now
    let x;
    assume
A1: x in I;
    hence (f*F)/.x = f(*)(F/.x) by Def6
      .= ((I-->f)/.x)(*)(F/.x) by A1,Th2
      .= ((I-->f)"*"F)/.x by A1,Def7;
  end;
  hence thesis by Th1;
end;
