reserve A,A1,A2,B,B1,B2,C,O for Ordinal,
      R,S for Relation,
      a,b,c,o,l,r for object;
reserve x,y,z,t,r,l for Surreal,
        X,Y,Z for set;

theorem ::: move to preliminaries
  for X1,X2,Y1,Y2 be set st
    (for x st x in X1 ex y st y in X2 & x <= y) &
    (for x st x in Y2 ex y st y in Y1 & y <= x) &
     x = [X1,Y1] & y = [X2,Y2]
  holds x <= y
proof
  let X1,X2,Y1,Y2 be set such that
  A1: (for x st x in X1 ex y st y in X2 & x <= y) &
  (for x st x in Y2 ex y st y in Y1 & y <= x) & x = [X1,Y1] & y = [X2,Y2];
  A2: L_x << {y}
  proof
    assume not L_x << {y};
    then consider l,r such that
    A3:  l in L_x & r in {y} & r <= l;
    consider z such that
    A4: z in L_y & l <=z by A3,A1;
    r=y by A3,TARSKI:def 1;
    then L_y << {l} & l in {l} by TARSKI:def 1,A3,Th43;
    hence thesis by A4;
  end;
  {x} << R_y
  proof
    assume not {x} << R_y;
    then consider l,r such that
    A5:l in {x} & r in R_y & r <= l;
    consider z such that
    A6: z in R_x and A7:  z <= r by A5,A1;
    l=x by TARSKI:def 1,A5;
    then r in {r} & {r} << R_x by TARSKI:def 1,A5,Th43;
    hence thesis by A6,A7;
  end;
  hence thesis by A2,Th43;
end;
