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 Th4:
  real-anti-diagonal is closed Subset of Sorgenfrey-plane
proof
     set L = real-anti-diagonal;
     set S2 = Sorgenfrey-plane;
A1:  L c= [#] S2
     proof
       let z be object;
       assume z in L;
       then consider x,y being Real such that
   A2:  z = [x,y] and y = -x;
        x in REAL & y in REAL by XREAL_0:def 1; then
        x in [#] Sorgenfrey-line & y in [#] Sorgenfrey-line by TOPGEN_3:def 2;
        then [x,y] in [:[#] Sorgenfrey-line, [#] Sorgenfrey-line:]
               by ZFMISC_1:def 2;
       hence z in [#] Sorgenfrey-plane by A2;
     end;
     reconsider L = real-anti-diagonal as Subset of Sorgenfrey-plane by A1;
     defpred Pf[object, object] means
     ex x,y being Real st $1 = [x,y] & $2 = x+y;
A3:  for z being object st z in the carrier of S2
       ex u being object st u in the carrier of R^1 & Pf[z,u]
     proof
       let z be object;
       assume z in the carrier of S2;
       then z in [:the carrier of Sorgenfrey-line,
          the carrier of Sorgenfrey-line:] by BORSUK_1:def 2;
        then consider x,y being object such that
   A4:  x in the carrier of Sorgenfrey-line and
   A5:  y in the carrier of Sorgenfrey-line and
   A6:  z = [x,y] by ZFMISC_1:def 2;
       reconsider x as Element of REAL by A4, TOPGEN_3:def 2;
       reconsider y as Element of REAL by A5, TOPGEN_3:def 2;
        set u = x+y;
        u in the carrier of R^1 by TOPMETR:17;
       hence thesis by A6;
     end;
    consider f being Function of S2, R^1 such that
A7:  for z being object st z in the carrier of S2 holds Pf[z,f.z]
        from FUNCT_2:sch 1(A3);
A8: for x,y being Element of REAL st [x,y] in the carrier of S2
     holds f.([x,y]) = x + y
     proof
       let x,y be Element of REAL such that
   A9:  [x,y] in the carrier of S2;
       consider x1,y1 being Real such that
   A10:  [x,y] = [x1,y1] & f.([x,y]) = x1+y1 by A9, A7;
        x = x1 & y = y1 by A10, XTUPLE_0:1;
       hence thesis by A10;
     end;
    for p being Point of S2, r being positive Real
     ex W being open Subset of S2 st p in W & f.:W c= ].f.p-r,f.p+r.[
    proof
      let p be Point of S2;
      let r be positive Real;
       p in [#] S2; then
       p in [:the carrier of Sorgenfrey-line, the carrier of Sorgenfrey-line:]
       by BORSUK_1:def 2;
       then consider x,y being object such that
  A11:  x in the carrier of Sorgenfrey-line and
  A12:  y in the carrier of Sorgenfrey-line and
  A13:  p = [x,y] by ZFMISC_1:def 2;
      reconsider x as Element of REAL by A11, TOPGEN_3:def 2;
      reconsider y as Element of REAL by A12, TOPGEN_3:def 2;
  A14:  f.p = x+y by A13, A8;
       set U = ].f.p-r,f.p+r.[;
       set W = [:[.x,x+(r/2).[ , [.y,y+(r/2).[:];
  A15:  W c= [#] S2
       proof
         let z be object;
         assume z in W; then
         consider u,v being object such that
     A16:  u in [.x,x+(r/2).[ & v in [.y,y+(r/2).[ and
     A17:  z = [u,v] by ZFMISC_1:def 2;
          reconsider u,v as Element of [#] Sorgenfrey-line
            by A16, TOPGEN_3:def 2;
          u in [#] Sorgenfrey-line & v in [#] Sorgenfrey-line; then
          z in [:[#] Sorgenfrey-line, [#] Sorgenfrey-line:] by
          A17, ZFMISC_1:def 2;
         hence z in [#] S2;
       end;
      reconsider W as Subset of S2 by A15;
      reconsider X = [.x,x+(r/2).[ as Subset of Sorgenfrey-line by
       TOPGEN_3:def 2;
      reconsider Y = [.y,y+(r/2).[ as Subset of Sorgenfrey-line by
       TOPGEN_3:def 2;
       X is open & Y is open by TOPGEN_3:11; then
  A18:  W is open by BORSUK_1:6;
       r/2 is positive;
       then x < x+(r/2) & y < y+(r/2) by XREAL_1:29; then
       x in [.x,x+(r/2).[ & y in [.y,y+(r/2).[ by XXREAL_1:3; then
  A19:  p in W by A13, ZFMISC_1:def 2;
       f.:W c= U
       proof
         let z be object;
         assume z in f.:W;
         then consider w being object such that
          w in dom f and
     A20:  w in W and
     A21:  z = f.w by FUNCT_1:def 6;
         consider x1,y1 being object such that
     A22:  x1 in X & y1 in Y and
     A23:  w = [x1,y1] by A20, ZFMISC_1:def 2;
         reconsider x1 as Element of REAL by A22;
         reconsider y1 as Element of REAL by A22;
     A24:  x <= x1 & x1 < x+(r/2) & y <= y1 & y1 < y+(r/2) by A22, XXREAL_1:3;
     A25:  x+y <= x1+y1 by XREAL_1:7, A24;
          x+y-r < x+y by XREAL_1:44,XXREAL_0:def 6; then
     A26:  x+y-r < x1+y1 by A25, XXREAL_0:2;
          x1+y1 < (x+(r/2))+(y+(r/2)) by XREAL_1:8, A24; then
          x1+y1 in U by A26, A14, XXREAL_1:4;
         hence z in U by A23, A8, A20, A21;
       end;
      hence thesis by A19, A18;
    end; then
A27: f is continuous by TOPS_4:21;
    reconsider zz = 0 as Element of REAL by XREAL_0:def 1;
    reconsider k = {zz} as Subset of R^1 by TOPMETR:17;
    reconsider k1 = [.zz, zz.] as Subset of R^1 by TOPMETR:17;
A28: k = k1 by XXREAL_1:17;
    L = f"k
    proof
      hereby
        let z be object; assume
    A29:  z in L; then
        consider x,y being Real such that
    A30:  z = [x,y] and
    A31:  y = -x;
        reconsider x,y as Element of REAL by XREAL_0:def 1;
         f.z = x + y by A8, A30, A29; then
        f.z in k by TARSKI:def 1, A31;
        hence z in f"k by FUNCT_2:38, A29;
      end;
        let z be object; assume z in f"k; then
A32:      z in [#] S2 & f.z in k by FUNCT_2:38; then
    A33: z in [:the carrier of Sorgenfrey-line, the carrier of Sorgenfrey-line
:]
         by BORSUK_1:def 2;
        consider x,y being object such that
    A34:  x in the carrier of Sorgenfrey-line and
    A35:  y in the carrier of Sorgenfrey-line and
    A36:  z = [x,y] by A33, ZFMISC_1:def 2;
        reconsider x1 = x as Element of REAL by A34, TOPGEN_3:def 2;
        reconsider y1 = y as Element of REAL by A35, TOPGEN_3:def 2;
        f.z = x1 + y1 by A8, A36, A32; then
        (x1 + y1) = 0 by A32, TARSKI:def 1;
        then -x1 = y1;
        hence z in L by A36;
    end;
   hence thesis by A28, A27, TREAL_1:1;
end;
