reserve fi,psi for Ordinal-Sequence,
  A,A1,B,C,D for Ordinal,
  X,Y for set,
  x,y for object;

theorem
  A*^B = 1 implies A = 1 & B = 1
proof
  assume
A1: A*^B = 1;
  then
A2: B <> {} by ORDINAL2:38;
  {} c= B;
  then {} c< B by A2;
  then {} in B by ORDINAL1:11;
  then
A3: 1 c= B by Lm1,ORDINAL1:21;
A4: now
A5: B = 1*^B by ORDINAL2:39;
    assume 1 in A;
    hence contradiction by A1,A2,A3,A5,ORDINAL1:5,ORDINAL2:40;
  end;
  now
    assume A in 1;
    then A c= {} by Lm1,ORDINAL1:22;
    then A = {} by XBOOLE_1:3;
    hence contradiction by A1,ORDINAL2:35;
  end;
  hence A = 1 by A4,ORDINAL1:14;
  hence thesis by A1,ORDINAL2:39;
end;
