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 Th3:
  card real-anti-diagonal = continuum
proof
    REAL, real-anti-diagonal are_equipotent
    proof
      defpred P[object, object] means
       ex x being Real st $1 = x & $2 = [x,-x];
  A1:  for r being object st r in REAL
         ex a being object st a in real-anti-diagonal & P[r,a]
       proof
         let r be object;
         assume r in REAL;
         then reconsider r as Real;
          set a = [r,-r];
          a in real-anti-diagonal;
         hence thesis;
       end;
      consider Z being Function of REAL, real-anti-diagonal such that
  A2:  for r being object st r in REAL holds P[r,Z.r] from FUNCT_2: sch 1(A1);
      take Z;
  A3:  real-anti-diagonal <> {}
       proof
         reconsider x = 1, y = -1 as Element of REAL by XREAL_0:def 1;
         set z = [x,y];
         z in real-anti-diagonal;
         hence real-anti-diagonal <> {};
       end;
  thus Z is one-to-one
       proof
         let r1,r2 be object such that
     A4:  r1 in dom Z & r2 in dom Z and
     A5:  Z.r1 = Z.r2;
        consider x1 being Real such that
     A6:  r1 = x1 & Z.r1 = [x1,-x1] by A2, A4;
        consider x2 being Real such that
     A7:  r2 = x2 & Z.r2 = [x2,-x2] by A2, A4;
        thus r1 = r2 by A6, A7, A5, XTUPLE_0:1;
       end;
  thus dom Z = REAL by A3, FUNCT_2:def 1;
  thus rng Z = real-anti-diagonal
       proof
         thus rng Z c= real-anti-diagonal;
         thus real-anti-diagonal c= rng Z
         proof
           let z be object;
           assume z in real-anti-diagonal; then
           consider x,y being Real such that
       A8:  z = [x,y] and
       A9:  y = -x;
           consider x1 being Real such that
       A10:  x = x1 & Z.x = [x1,-x1] by A2,XREAL_0:def 1;
          thus z in rng Z by A10, A8, A9, FUNCT_2:4, A3,XREAL_0:def 1;
         end;
       end;
  end;
  hence card real-anti-diagonal = continuum by TOPGEN_3:def 4,CARD_1:5;
end;
