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

theorem
   (x+y)*(x+y) == x*x + y*y +(x*y + y*x)
proof
  x*(x+y) == x*x + x*y & y*(x+y) == y*x + y*y by Th67;
  then A1: (x+y)*(x+y) == x*(x+y) + y*(x+y) == (x*x+x*y) + (y*x + y*y)
  by Th67,Th43;
  (x*x+x*y) + (y*x + y*y) = x*x+(x*y + (y*x + y*y)) by Th37
  .= x*x+(x*y + y*x + y*y) by Th37
  .= x*x+y*y+(x*y + y*x) by Th37;
  hence thesis by A1,SURREALO:4;
end;
