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

theorem Th20:
  for X1,X2 be set holds
    --(X1\/X2)  = (-- X1) \/ (--X2)
proof
  let X1,X2 be set;
  thus --(X1\/X2)  c= (-- X1) \/ (--X2)
  proof
    let xy be object;
    assume xy in --(X1\/X2);
    then consider x be Surreal such that
    A1:  x in X1\/X2 & xy= -x by Def4;
    x in X1 or x in X2 by A1,XBOOLE_0:def 3;
    then xy in (--X1) or xy in (--X2) by A1,Def4;
    hence thesis by XBOOLE_0:def 3;
  end;
  let xy be object;
  assume xy in (-- X1) \/ (--X2);
  then per cases by XBOOLE_0:def 3;
  suppose xy in --X1;
    then consider x be Surreal such that
    A2:  x in X1  & xy = -x by Def4;
    x in X1\/X2 by A2,XBOOLE_0:def 3;
    hence thesis by A2,Def4;
  end;
  suppose xy in --X2;
    then consider x be Surreal such that
    A3:  x in X2  & xy=-x by Def4;
    x in X1\/X2 by A3,XBOOLE_0:def 3;
    hence thesis by A3,Def4;
  end;
end;
