theorem
  A9 is closed Ideal of X9 implies f"A9 is closed Ideal of X
proof
  assume
A1: A9 is closed Ideal of X9;
  then reconsider XY=f"A9 as Ideal of X by Th45;
  for y being Element of XY holds y` in XY
  proof
    let y be Element of XY;
A2: f.y in A9 by FUNCT_2:38;
    reconsider y as Element of X;
    (f.y)` in A9 by A1,A2,BCIALG_1:def 19;
    then f.0.X\f.y in A9 by Th35;
    then f.(y`) in A9 by Def6;
    hence thesis by FUNCT_2:38;
  end;
  hence thesis by BCIALG_1:def 19;
end;
