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

theorem Th14:
  L_y << {x} << R_y
    implies [L_x \/ L_y, R_x \/ R_y] is Surreal
proof
  assume A1: L_y << {x} << R_y;
  consider A be Ordinal such that A2: x in Day A by SURREAL0:def 14;
  consider B be Ordinal such that A3: y in Day B by SURREAL0:def 14;
  set X=L_x \/ L_y, Y = R_x \/ R_y;
  A4: x=[L_x,R_x]; then
  A5:L_x << R_x by A2,SURREAL0:46;
  A6: y=[L_y,R_y]; then
  A7: L_y << R_y by A3,SURREAL0:46;
  A8: x in {x} by TARSKI:def 1;
  A9: X << Y
  proof
    let x1,y1 be Surreal;
    assume x1 in X & y1 in Y;
    then per cases by XBOOLE_0:def 3;
    suppose x1 in L_x & y1 in R_x;
      hence thesis by A5;
    end;
    suppose x1 in L_y & y1 in R_y;
      hence thesis by A7;
    end;
    suppose A10: x1 in L_x & y1 in R_y;
      then A11: x <= y1 & not x >= y1 by A8,A1;
      L_x << {x} by Th11;
      hence thesis by A8,A10,A11,Th4;
    end;
    suppose A12:x1 in L_y & y1 in R_x;
      then A13: x1 <= x & not x1 >= x by A8,A1;
      {x} << R_x by Th11;
      hence thesis by A8,A12,A13,Th4;
    end;
  end;
  for x be object st x in X \/ Y ex O be Ordinal st O in A\/B & x in Day O
  proof
    let z be object;
    assume z in X \/ Y;
    then z in X or z in Y by XBOOLE_0:def 3;
    then per cases by XBOOLE_0:def 3;
    suppose z in L_x or z in R_x;
      then z in L_x \/ R_x by XBOOLE_0:def 3;
      then consider O be Ordinal such that
      A14: O in A & z in Day O by A4,A2,SURREAL0:46;
      O in A\/B by A14,XBOOLE_0:def 3;
      hence thesis by A14;
    end;
    suppose z in L_y or z in R_y;
      then z in L_y \/ R_y by XBOOLE_0:def 3;
      then consider O be Ordinal such that
      A15: O in B & z in Day O by A3,A6,SURREAL0:46;
      O in A\/B by A15,XBOOLE_0:def 3;
      hence thesis by A15;
    end;
  end;
  then [X,Y] in Day(A\/B) by SURREAL0:46,A9;
  hence thesis;
end;
