
theorem Th15:
  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, r being Element of
  DOM, p being Point of T st (Thunder G).p < r holds p in (Tempest G).r
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;
  let r be Element of DOM;
  let p be Point of T;
  assume
A2: (Thunder G).p < r;
  now
    per cases;
    suppose
A3:   Rainbow(p,G) = {};
      assume
A4:   not p in (Tempest G).r;
      r in halfline 0 \/ DYADIC or r in right_open_halfline 1 by URYSOHN1:def 3
,XBOOLE_0:def 3;
      then
A5:   r in halfline 0 or r in DYADIC or r in right_open_halfline 1 by
XBOOLE_0:def 3;
A6:   0 < r by A2,A3,Def6;
      now
        per cases by A6,A5,XXREAL_1:233;
        suppose
A7:       r in DYADIC;
          reconsider r1 = r as R_eal by XXREAL_0:def 1;
A8:       for s being Real st s = r1 holds not p in (Tempest G).s by A4;
          then reconsider
          S = Rainbow(p,G) as non empty Subset of ExtREAL by A7,Def5;
A9:       (Thunder G).p = sup S by Def6;
          r1 in Rainbow(p,G) by A7,A8,Def5;
          hence thesis by A2,A9,XXREAL_2:4;
        end;
        suppose
          r in right_open_halfline 1;
          then (Tempest G).r = the carrier of T by A1,Def4;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
    suppose
      Rainbow(p,G) <> {};
      then reconsider S = Rainbow(p,G) as non empty Subset of ExtREAL;
      reconsider e1 = 1 as R_eal by XXREAL_0:def 1;
      consider s being object such that
A10:  s in S by XBOOLE_0:def 1;
      reconsider s as R_eal by A10;
A11:  s <= sup S by A10,XXREAL_2:4;
      assume
A12:  not p in (Tempest G).r;
      r in halfline 0 \/ DYADIC or r in right_open_halfline 1 by URYSOHN1:def 3
,XBOOLE_0:def 3;
      then
A13:  r in halfline 0 or r in DYADIC or r in right_open_halfline 1 by
XBOOLE_0:def 3;
A14:  (Thunder G).p = sup S by Def6;
      for x being R_eal st x in S holds 0. <= x & x <= e1
      proof
        let x be R_eal;
        assume x in S;
        then
A15:    x in DYADIC by Def5;
        then reconsider x as Real;
        ex n being Nat st x in dyadic(n) by A15,URYSOHN1:def 2;
        hence thesis by URYSOHN1:1;
      end;
      then
A16:  0. <= s by A10;
      now
        per cases by A2,A14,A16,A11,A13,XXREAL_1:233;
        suppose
A17:      r in DYADIC;
          reconsider r1 = r as R_eal by XXREAL_0:def 1;
          for s being Real st s = r1 holds not p in (Tempest G).s by A12;
          then r1 in Rainbow(p,G) by A17,Def5;
          hence thesis by A2,A14,XXREAL_2:4;
        end;
        suppose
          r in right_open_halfline 1;
          then (Tempest G).r = the carrier of T by A1,Def4;
          hence thesis;
        end;
      end;
      hence thesis;
    end;
  end;
  hence thesis;
end;
