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 holds -- (X1++X2) = (--X1) ++ (--X2)
proof
   let X1,X2 be set;
   thus -- (X1++X2) c= (--X1) ++ (--X2)
   proof
     let a be object;
     assume a in -- (X1++X2);
     then consider xy be Surreal such that
     A1: xy in X1++X2 & a=-xy by Def4;
     consider x,y such that
     A2:  x in X1 & y in X2 & xy = x+y by A1,Def8;
     -x in --X1 & -y in --X2 by A2,Def4;
     then -(x+y) = -x + -y in (--X1) ++ (--X2) by Th40,Def8;
     hence thesis by A1,A2;
   end;
   let a be object;
   assume a in (--X1) ++ (--X2);
   then consider x1,y1 be Surreal such that
   A3: x1 in --X1 & y1 in --X2 & a = x1 + y1 by Def8;
   consider x such that
   A4: x in X1 & x1 = -x by A3,Def4;
   consider y such that
   A5: y in X2 & y1 = -y by A3,Def4;
   x+y in X1++X2 by A4,A5,Def8;
   then -x + -y = -(x+y) in -- (X1++X2) by Th40,Def4;
   hence thesis by A3,A4,A5;
end;
