reserve A,B,C for Ordinal,
        o for object,
        x,y,z,t,r,l for Surreal,
        X,Y for set;

theorem
  for X1,X2 be set st X1 <=_ X2 holds -- X1 <=_ -- X2
proof
  let X1,X2 be set such that A1:X1 <=_ X2;
  let x such that A2: x in -- X1;
  consider y such that
  A3:y in X1 & x = -y by A2,Def4;
  consider y2,y3 be Surreal such that
  A4:y2 in X2 & y3 in X2 & y2 <= y <= y3 by A3,A1;
  A5:-y2 in --X2 & -y3 in --X2 by A4,Def4;
  -y3 <= -y <= -y2 by Th10,A4;
  hence thesis by A5,A3;
end;
