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

theorem
  x==y & z = [L_x\/L_y,R_x\/R_y] implies x==z
proof
  assume A1: x==y & z = [L_x\/L_y,R_x\/R_y];
  L_y << {y} << R_y by Th11;
  then L_y << {x} << R_y by A1,Th17,Th18;
  hence thesis by A1,Th15;
end;
