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,Y1,Y2 be set st X1 <=_ X2 & Y1 <=_ Y2
    holds X1++Y1 <=_ X2++Y2
proof
   let X1,X2,Y1,Y2 be set such that
   A1: X1 <=_ X2 & Y1 <=_ Y2;
   let x such that A2: x in X1++Y1;
   consider x1,y1 be Surreal such that
   A3:x1 in X1 & y1 in Y1 & x = x1 + y1 by A2,Def8;
   consider x2,x3 be Surreal such that
   A4:x2 in X2 & x3 in X2 & x2 <= x1 <= x3 by A3,A1;
   consider y2,y3 be Surreal such that
   A5:y2 in Y2 & y3 in Y2 & y2 <= y1 <= y3 by A3,A1;
   take z2=x2+y2,z3=x3+y3;
   x2+y2 <= x1+y2 <= x1 +y1 <= x3+y1 <= x3+y3 by A4,A5,Th32;
   hence thesis by A4,A5,Def8,A3,SURREALO:4;
end;
