reserve T for TopSpace,
  x, y, a, b, U, Ux, rx for set,
  p, q for Rational,
  F, G for Subset-Family of T,
  Us, I for Subset-Family of Sorgenfrey-line;

theorem Th2:
  [:RAT,RAT:] is dense Subset of Sorgenfrey-plane
proof
    [:RAT, RAT:] c= [#] Sorgenfrey-plane
    proof
      let z be object;
      assume z in [:RAT, RAT:]; then
      consider x,y being object such that
  A1:  x in RAT & y in RAT and
  A2:  z = [x,y] by ZFMISC_1:def 2;
       x in REAL & y in REAL by A1, NUMBERS:12; then
       x in [#] Sorgenfrey-line &
       y in [#] Sorgenfrey-line by TOPGEN_3:def 2; then
       z in [:[#] Sorgenfrey-line, [#] Sorgenfrey-line:]
       by A2, ZFMISC_1:def 2;
      hence z in [#] Sorgenfrey-plane;
    end;
    then reconsider C = [:RAT, RAT:] as Subset of Sorgenfrey-plane;
    for A being Subset of Sorgenfrey-plane st A <> {} & A is open
      holds A meets C
    proof
      let A be Subset of Sorgenfrey-plane such that
  A3:  A <> {} and
  A4:  A is open;
      consider B being Subset-Family of Sorgenfrey-plane such that
  A5:  A = union B and
  A6:  (for e being set st e in B holds ex X1 being Subset of Sorgenfrey-line,
       Y1 being Subset of Sorgenfrey-line st (e = [:X1,Y1:] & X1 is open &
       Y1 is open)) by BORSUK_1:5, A4;
      now
        assume A7: for e being set st e in B holds e = {};
         union B c= {}
         proof
           let z be object;
           assume z in union B; then
           consider y being set such that
       A8:  z in y & y in B by TARSKI:def 4;
           thus z in {} by A8, A7;
         end;
        hence contradiction by A5, A3;
      end;
      then
      consider e being set such that
  A9:  e in B & e <> {};
      consider X1,Y1 being Subset of Sorgenfrey-line such that
  A10:  e = [:X1,Y1:] and
  A11:  X1 is open & Y1 is open by A6, A9;
  A12:  X1 <> {} & Y1 <> {} by ZFMISC_1:90, A9, A10;
      consider x1 being object such that
  A13:  x1 in X1 by A12, XBOOLE_0:7;
      consider y1 being object such that
  A14:  y1 in Y1 by A12, XBOOLE_0:7;
      consider ax being Subset of Sorgenfrey-line such that
  A15:  ax in BB and
       x1 in ax and
  A16:  ax c= X1 by YELLOW_9:31, A11, A13, Lm8;
      consider ay being Subset of Sorgenfrey-line such that
  A17:  ay in BB and
       y1 in ay and
  A18:  ay c= Y1 by YELLOW_9:31, A11, A14, Lm8;
      consider px,qx being Real such that
  A19:  ax = [.px,qx.[ and
  A20:  px < qx & qx is rational by A15, Lm7;
      consider py,qy being Real such that
  A21:  ay = [.py,qy.[ and
  A22:  py < qy & qy is rational by A17, Lm7;
      consider rx being Rational such that
  A23:  px < rx & rx < qx by RAT_1:7, A20;
  A24:  rx in ax by A23, XXREAL_1:3, A19;
      consider ry being Rational such that
  A25:  py < ry & ry < qy by RAT_1:7, A22;
  A26:  ry in ay by A25, XXREAL_1:3, A21;
       rx in RAT & ry in RAT by RAT_1:def 2; then
  A27:  [rx,ry] in C by ZFMISC_1:def 2;
       [rx,ry] in [:X1,Y1:] by A24, A26, A16, A18, ZFMISC_1:def 2; then
  A28:  [rx,ry] in A by A5, A9, A10, TARSKI:def 4;
      thus A meets C by A27, A28, XBOOLE_0:3;
    end;
   hence [:RAT, RAT:] is dense Subset of Sorgenfrey-plane by TOPS_1:45;
end;
