
theorem Th11:
  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 Real, C
  being Subset of T st C = (Tempest G).r & r in DOM holds C is open
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 Real;
  let C be Subset of T;
  assume that
A2: C = (Tempest G).r and
A3: r in DOM;
A4: r in halfline 0 \/ DYADIC or r in right_open_halfline 1 by A3,
URYSOHN1:def 3,XBOOLE_0:def 3;
  per cases by A4,XBOOLE_0:def 3;
  suppose
    r in halfline 0;
    then C = {} by A1,A2,A3,Def4;
    then C in the topology of T by PRE_TOPC:1;
    hence thesis;
  end;
  suppose
A5: r in DYADIC;
    then consider n being Nat such that
A6: r in dyadic(n) by URYSOHN1:def 2;
    reconsider D = G.n as Drizzle of A,B,n by A1,Def2;
A7: for r1,r2 being Element of dyadic(n) st r1 < r2 holds D.r1 is open &
    D.r2 is open & Cl(D.r1) c= D.r2 & A c= D.0 & B = [#]T \ D.1 by A1,Def1;
A8: (Tempest G).r = (G.n).r by A1,A3,A5,A6,Def4;
    now
      per cases by A6,URYSOHN1:1;
      case
A9:     r = 0;
        then reconsider r as Element of dyadic(n) by URYSOHN1:6;
        1 is Element of dyadic(n) & D.r = C by A1,A2,A3,A5,Def4,URYSOHN1:6;
        hence thesis by A1,A9,Def1;
      end;
      case
A10:    0 < r;
        0 in dyadic(n) by URYSOHN1:6;
        hence thesis by A2,A6,A8,A7,A10;
      end;
    end;
    hence thesis;
  end;
  suppose
A11: r in right_open_halfline 1;
A12: the carrier of T in the topology of T by PRE_TOPC:def 1;
    C = the carrier of T by A1,A2,A3,A11,Def4;
    hence thesis by A12;
  end;
end;
