theorem
  f is onto implies f.:CI is closed Ideal of X9
proof
  assume f is onto;
  then reconsider Kf = f.:CI as Ideal of X9 by Th47;
  now
    let x9 be Element of Kf;
    consider x being object such that
    x in dom f and
A1: x in CI and
A2: x9 = f.x by FUNCT_1:def 6;
    reconsider x as Element of CI by A1;
    x` in the carrier of X;
    then x` in dom f by FUNCT_2:def 1;
    then x` in CI & [x`,f.(x`)] in f by BCIALG_1:def 19,FUNCT_1:1;
    then f.(x`)in f.:CI by RELAT_1:def 13;
    then f.(0.X)\f.x in f.:CI by Def6;
    hence x9` in Kf by A2,Th35;
  end;
  hence thesis by BCIALG_1:def 19;
end;
