reserve m for Cardinal,
  A,B,C for Ordinal,
  x,y,z,X,Y,Z,W for set,
  f for Function;

theorem Th9:
  W is Tarski implies On W = card W
proof
  assume
A1: W is Tarski;
  now
    let X;
    assume
A2: X in On W;
    hence X is Ordinal by ORDINAL1:def 9;
    reconsider A = X as Ordinal by A2,ORDINAL1:def 9;
A3: X in W by A2,ORDINAL1:def 9;
    thus X c= On W
    proof
      let x be object;
      assume
A4:   x in X;
      then x in A;
      then reconsider B = x as Ordinal;
      B c= A by A4,ORDINAL1:def 2;
      then B in W by A1,A3,CLASSES1:def 1;
      hence thesis by ORDINAL1:def 9;
    end;
  end;
  then reconsider ON = On W as epsilon-transitive epsilon-connected set
   by ORDINAL1:19;
A5: now
    assume ON in W;
    then ON in ON by ORDINAL1:def 9;
    hence contradiction;
  end;
  ON c= W by ORDINAL2:7;
  then
A6: ON,W are_equipotent or ON in W by A1;
  now
    let A;
    assume that
A7: A,ON are_equipotent and
A8: not ON c= A;
    A in ON by A8,ORDINAL1:16;
    then A in W by ORDINAL1:def 9;
    hence contradiction by A1,A6,A5,A7,Th1,WELLORD2:15;
  end;
  then reconsider ON as Cardinal by CARD_1:def 1;
  ON = card ON;
  hence thesis by A6,A5,CARD_1:5;
end;
