
theorem Th10:
  for T being non empty normal TopSpace, A,B being closed Subset
  of T st A <> {} & A misses B holds for G being Rain of A,B holds ex F being
Function of DOM,bool the carrier of T st for x being Real st x in DOM holds (x
in halfline 0 implies F.x = {}) & (x in right_open_halfline 1 implies F.x = the
carrier of T) & (x in DYADIC implies for n being Nat st x in dyadic(
  n) holds F.x = (G.n).x )
proof
  let T be non empty normal TopSpace;
  let A,B be closed Subset of T;
  assume
A1: A <> {} & A misses B;
  let G be Rain of A,B;
  defpred P[Element of DOM,set] means (($1 in halfline 0 implies $2 = {}) & (
  $1 in right_open_halfline 1 implies $2 = the carrier of T) & ($1 in DYADIC
implies (for n being Nat st $1 in dyadic(n) holds $2 = (G.n).$1)));
A2: for x being Element of DOM ex y being Subset of T st P[x,y]
  proof
    reconsider a = 0, b = 1 as R_eal by XXREAL_0:def 1;
    let x be Element of DOM;
A3: x in halfline 0 \/ DYADIC or x in right_open_halfline 1 by URYSOHN1:def 3
,XBOOLE_0:def 3;
    per cases by A3,XBOOLE_0:def 3;
    suppose
A4:   x in halfline 0;
A5:   not x in right_open_halfline 1 & not x in DYADIC
      proof
        assume
A6:     x in right_open_halfline 1 or x in DYADIC;
        per cases by A6;
        suppose
          x in right_open_halfline 1;
          then 1 < x by XXREAL_1:235;
          hence thesis by A4,XXREAL_1:233;
        end;
        suppose
A7:       x in DYADIC;
          reconsider x as R_eal by XXREAL_0:def 1;
          a <= x by A7,URYSOHN2:28,XXREAL_1:1;
          hence thesis by A4,XXREAL_1:233;
        end;
      end;
      reconsider s = {} as Subset of T by XBOOLE_1:2;
      s = s;
      hence thesis by A5;
    end;
    suppose
A8:   x in DYADIC;
A9:   not x in halfline 0
      proof
        assume
A10:    x in halfline 0;
        reconsider x as R_eal by XXREAL_0:def 1;
        a <= x by A8,URYSOHN2:28,XXREAL_1:1;
        hence thesis by A10,XXREAL_1:233;
      end;
A11:  not x in right_open_halfline 1
      proof
        assume
A12:    x in right_open_halfline 1;
        reconsider x as R_eal by XXREAL_0:def 1;
        x <= b by A8,URYSOHN2:28,XXREAL_1:1;
        hence thesis by A12,XXREAL_1:235;
      end;
      ex s being Subset of T st for n being Nat st x in dyadic(
      n) holds s = (G.n).x by A1,A8,Th9;
      hence thesis by A11,A9;
    end;
    suppose
A13:  x in right_open_halfline 1;
A14:  not x in halfline 0 & not x in DYADIC
      proof
        assume
A15:    x in halfline 0 or x in DYADIC;
        per cases by A15;
        suppose
          x in halfline 0;
          then x < 0 by XXREAL_1:233;
          hence thesis by A13,XXREAL_1:235;
        end;
        suppose
A16:      x in DYADIC;
          reconsider x as R_eal by XXREAL_0:def 1;
          x <= b by A16,URYSOHN2:28,XXREAL_1:1;
          hence thesis by A13,XXREAL_1:235;
        end;
      end;
      the carrier of T c= the carrier of T;
      then reconsider s = the carrier of T as Subset of T;
      s = s;
      hence thesis by A14;
    end;
  end;
  ex F being Function of DOM,bool the carrier of T st for x being Element
  of DOM holds P[x,F.x] from FUNCT_2:sch 3(A2);
  then consider F being Function of DOM,bool the carrier of T such that
A17: for x being Element of DOM holds P[x,F.x];
  take F;
  thus thesis by A17;
end;
