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 Th5:
  for A being Subset of Sorgenfrey-plane st A = real-anti-diagonal
    holds Der A is empty
proof
    let A be Subset of Sorgenfrey-plane such that
A1: A = real-anti-diagonal;
    assume Der A is not empty; then
    consider a being object such that
A2:  a in Der A by XBOOLE_0:7;
     a is_an_accumulation_point_of A by TOPGEN_1:def 3, A2; then
A3:  a in Cl(A\{a}) by TOPGEN_1:def 2;
    consider x,y being object such that
A4:  x in REAL & y in REAL and
A5:  a = [x,y] by ZFMISC_1:def 2, Lm10, A2;
    reconsider x,y as Real by A4;
    per cases;
     suppose
   A6:  y >= -x;
        set Gx = [.x,x+1.[;
        set Gy = [.y,y+1.[;
       reconsider Gx,Gy as Subset of Sorgenfrey-line by TOPGEN_3:def 2;
        set G = [:Gx,Gy:];
       reconsider G as Subset of Sorgenfrey-plane;
        Gx is open & Gy is open by TOPGEN_3:11; then
   A7:  G is open by BORSUK_1:6;
           x <= x & x < x+1 by XREAL_1:29; then
   A8:  x in Gx by XXREAL_1:3;
        y <= y & y < y+1 by XREAL_1:29; then
        y in Gy by XXREAL_1:3; then
   A9:  a in G by A5, A8, ZFMISC_1:def 2;
          now
            assume G /\ (A\{a}) <> {};
            then consider z being object such that
        A10:  z in G /\ (A\{a}) by XBOOLE_0:def 1;
             z in G & z in (A\{a}) by XBOOLE_0:def 4, A10; then
        A11:  z in G & z in A & not z in {a} by XBOOLE_0:def 5;
            consider xz,yz being object such that
        A12:  xz in [.x,x+1.[ & yz in [.y,y+1.[ and
        A13:  z = [xz,yz] by ZFMISC_1:def 2, A11;
            reconsider xz as Real by A12;
            reconsider yz as Real by A12;
        A14:  x <= xz & y<= yz by A12, XXREAL_1:3;
        A15:  -x >= -xz & y <= -xz by A14, Lm11, A1, A11, A13, XREAL_1:24;
        A16:  -x >= y by XXREAL_0:2, A15;
        A17:  -x = y by A16, A6, XXREAL_0:1; then
        A18:  -x <= yz by A12, XXREAL_1:3;
        A19:  -x <= -xz by Lm11, A1, A11, A13, A18;
        A20:  x >= xz by XREAL_1:24, A19;
        A21:  x = xz by A20, A14, XXREAL_0:1; then
             y = yz by A17, Lm11, A1, A11, A13;
            hence contradiction by A11, TARSKI:def 1, A13, A5, A21;
          end; then
         G misses (A\{a}) by XBOOLE_0:def 7;
        hence contradiction by A7, A9, A3, PRE_TOPC:def 7;
      end;
      suppose
    A22:  y < -x;
         set Gx = [.x,x+(|.x+y.|/2).[;
         set Gy = [.y,y+(|.x+y.|/2).[;
        reconsider Gx, Gy as Subset of Sorgenfrey-line by TOPGEN_3:def 2;
        reconsider G = [:Gx,Gy:] as Subset of Sorgenfrey-plane;
         Gx is open & Gy is open by TOPGEN_3:11; then
    A23:  G is open by BORSUK_1:6;
    A24:  y + x < 0 by A22, XREAL_1:61;
    A25:  |.x+y.| = -(x+y) by ABSVALUE:def 1,A22, XREAL_1:61;
A26:      |.x+y.| > 0 by A24, A25,XREAL_1:58;
         then x < x+(|.x+y.|/2) by XREAL_1:29,139; then
    A27:  x in Gx by XXREAL_1:3;
         y < y+(|.x+y.|/2) by XREAL_1:29, A26, XREAL_1:139; then
         y in Gy by XXREAL_1:3; then
    A28:  a in G by A5, A27, ZFMISC_1:def 2;
          now
            assume G /\ (A\{a}) <> {};
            then consider z being object such that
         A29:  z in G /\ (A\{a}) by XBOOLE_0:def 1;
              z in G & z in (A\{a}) by A29, XBOOLE_0:def 4; then
         A30:  z in G & z in A & not z in {a} by XBOOLE_0:def 5;
             consider xz,yz being object such that
         A31:  xz in [.x,x+(|.x+y.|/2).[ & yz in [.y,y+(|.x+y.|/2).[ and
         A32:  z = [xz,yz] by ZFMISC_1:def 2, A30;
             reconsider xz,yz as Real by A31;
         A33:  yz = -xz by Lm11, A1, A30, A32;
         A34:  xz < x+(|.x+y.|/2) & yz < y+(|.x+y.|/2) by A31, XXREAL_1:3;
              xz + yz < (x+(|.x+y.|/2)) + (y+(|.x+y.|/2)) by A34, XREAL_1:8;
             hence contradiction by A33, A25;
           end; then
         G misses (A\{a}) by XBOOLE_0:def 7;
        hence contradiction by A23, A28, A3, PRE_TOPC:def 7;
      end;
end;
