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

theorem Th13:
  born_eq x = born_eq -x
proof
  consider y such that
  A1: born y = born_eq x & y ==x by SURREALO:def 5;
  A2: -y == - x by A1,Th10;
  A3: born y = born -y by Th12;
  for z st z == -x holds born y c= born z
  proof
    let z;
    assume z == -x;
    then -z == - -x = x by Th10;
    then born_eq x c= born -z by SURREALO:def 5;
    hence thesis by A1,Th12;
  end;
  hence thesis by A1,A2,A3,SURREALO:def 5;
end;
