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

theorem Th36:
  {x}++{y}={x+y}
proof
  thus {x}++{y}c={x+y}
  proof
    let xy be object;
    assume xy in {x}++{y};
    then consider a,b be Surreal such that
    A1:a in {x} & b in {y} & xy=a+b by Def8;
    a=x & b=y by A1,TARSKI:def 1;
    hence thesis by A1,TARSKI:def 1;
  end;
  A2:x in {x} & y in {y} by TARSKI:def 1;
  o in {x+y} implies o in {x}++{y}
  proof
  assume o in {x+y};
  then o=x+y by TARSKI:def 1;
  hence thesis by A2,Def8;
  end;
  hence thesis by TARSKI:def 3;
end;
