
theorem Th12:
  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 for r1,r2
being Real st r1 in DOM & r2 in DOM & r1 < r2 holds for C being Subset of T st
  C = (Tempest G).r1 holds Cl C c= (Tempest G).r2
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 r1,r2 be Real;
  assume that
A2: r1 in DOM and
A3: r2 in DOM and
A4: r1 < r2;
A5: r1 in halfline 0 \/ DYADIC or r1 in right_open_halfline 1 by A2,
URYSOHN1:def 3,XBOOLE_0:def 3;
A6: r2 in halfline 0 \/ DYADIC or r2 in right_open_halfline 1 by A3,
URYSOHN1:def 3,XBOOLE_0:def 3;
  let C be Subset of T;
  assume
A7: C = (Tempest G).r1;
  per cases by A5,A6,XBOOLE_0:def 3;
  suppose
A8: r1 in halfline 0 & r2 in halfline 0;
    C = {} by A1,A2,A7,A8,Def4;
    then Cl C = {} by PRE_TOPC:22;
    hence thesis;
  end;
  suppose
    r1 in DYADIC & r2 in halfline 0;
    then r2 < 0 & ex s being Nat st r1 in dyadic(s) by
URYSOHN1:def 2,XXREAL_1:233;
    hence thesis by A4,URYSOHN1:1;
  end;
  suppose
A9: r1 in right_open_halfline 1 & r2 in halfline 0;
    then 1 < r1 by XXREAL_1:235;
    hence thesis by A4,A9,XXREAL_1:233;
  end;
  suppose
A10: r1 in halfline 0 & r2 in DYADIC;
    C = {} by A1,A2,A7,A10,Def4;
    then Cl C = {} by PRE_TOPC:22;
    hence thesis;
  end;
  suppose
A11: r1 in DYADIC & r2 in DYADIC;
    then consider n2 being Nat such that
A12: r2 in dyadic(n2) by URYSOHN1:def 2;
    consider n1 being Nat such that
A13: r1 in dyadic(n1) by A11,URYSOHN1:def 2;
    set n = n1 + n2;
A14: dyadic(n1) c= dyadic(n) by NAT_1:11,URYSOHN2:29;
    then
A15: (Tempest G).r1 = (G.n).r1 by A1,A2,A11,A13,Def4;
    dyadic(n2) c= dyadic(n) by NAT_1:11,URYSOHN2:29;
    then reconsider r1,r2 as Element of dyadic(n) by A13,A12,A14;
    reconsider D = G.n as Drizzle of A,B,n by A1,Def2;
    Cl(D.r1) c= D.r2 by A1,A4,Def1;
    hence thesis by A1,A3,A7,A11,A15,Def4;
  end;
  suppose
A16: r1 in right_open_halfline 1 & r2 in DYADIC;
    then ex s being Nat st r2 in dyadic(s) by URYSOHN1:def 2;
    then
A17: r2 <= 1 by URYSOHN1:1;
    1 < r1 by A16,XXREAL_1:235;
    hence thesis by A4,A17,XXREAL_0:2;
  end;
  suppose
A18: r1 in halfline 0 & r2 in right_open_halfline 1;
    C = {} by A1,A2,A7,A18,Def4;
    then Cl C = {} by PRE_TOPC:22;
    hence thesis;
  end;
  suppose
    r1 in DYADIC & r2 in right_open_halfline 1;
    then (Tempest G).r2 = the carrier of T by A1,A3,Def4;
    hence thesis;
  end;
  suppose
    r1 in right_open_halfline 1 & r2 in right_open_halfline 1;
    then (Tempest G).r2 = the carrier of T by A1,A3,Def4;
    hence thesis;
  end;
end;
