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

theorem Th15:
  for X  be surreal-membered set
    holds -- -- X = X
proof
  let X  be surreal-membered set;
  thus -- -- X c= X
  proof
    let x be object;
    assume x in -- -- X;
    then consider y be Surreal such that A1: y in --X & x = -y by Def4;
    ex z be Surreal st z in X & y = -z by A1,Def4;
    hence thesis by A1;
  end;
  let x be object;
  assume A2: x in X;
  then reconsider x as Surreal by SURREAL0:def 16;
  -x in --X by A2,Def4;
  then  - -x in -- --X by Def4;
  hence thesis;
end;
