
theorem Th4:
  for T being non empty normal TopSpace, A,B being closed Subset of
  T st A <> {} & A misses B holds ex F being Functional_Sequence of DYADIC,bool
the carrier of T st for n being Nat holds F.n is Drizzle of A,B,n &
  for r being Element of dom (F.n) holds (F.n).r = (F.(n+1)).r
proof
  let T be non empty normal TopSpace;
  let A,B be closed Subset of T;
  defpred P[object] means ex n being Nat st $1 is Drizzle of A,B,n;
  consider D being set such that
A1: for x being object holds x in D iff x in PFuncs(DYADIC,bool the carrier
  of T) & P[x] from XBOOLE_0:sch 1;
  set S = the Drizzle of A,B,0;
A2: S is Element of PFuncs(DYADIC,bool the carrier of T) by Th3;
  then reconsider D as non empty set by A1;
  reconsider S as Element of D by A1,A2;
  defpred P1[Element of D,Element of D] means ex xx,yy being PartFunc of
  DYADIC,bool the carrier of T st (xx=$1 & yy = $2 & (ex k being Nat
st xx is Drizzle of A,B,k) & (for k being Nat st xx is Drizzle of A,
B,k holds yy is Drizzle of A,B,k+1) & (for r being Element of dom xx holds xx.r
  = yy.r));
  defpred Q[Nat,Element of D,Element of D] means P1[$2,$3];
  assume
A3: A <> {} & A misses B;
A4: for n being Nat for x being Element of D ex y being Element
  of D st Q[n,x,y]
  proof
    let n be Nat;
    let x be Element of D;
    consider s0 being Nat such that
A5: x is Drizzle of A,B,s0 by A1;
    reconsider xx = x as Drizzle of A,B,s0 by A5;
    consider yy being Drizzle of A,B,s0+1 such that
A6: for r being Element of dyadic(s0+1) holds (r in dyadic(s0) implies
    yy.r = xx.r) by A3,Th2;
A7: for r being Element of dom xx holds xx.r = yy.r
    proof
      let r be Element of dom xx;
      dom xx = dyadic(s0) by FUNCT_2:def 1;
      then
A8:   r in dyadic(s0);
      dyadic(s0) c= dyadic(s0+1) by URYSOHN1:5;
      hence thesis by A6,A8;
    end;
A9: for k being Nat st xx is Drizzle of A,B,k holds yy is
    Drizzle of A,B,k+1
    proof
      let k be Nat;
      assume xx is Drizzle of A,B,k;
      then
A10:  dom xx = dyadic(k) by FUNCT_2:def 1;
      k = s0
      proof
        assume
A11:    k <> s0;
        per cases by A11,XXREAL_0:1;
        suppose
A12:      k < s0;
          set o = 1/(2|^s0);
A13:      not o in dyadic(k)
          proof
A14:        (2|^k) < 1 * (2|^s0) by A12,PEPIN:66;
            assume o in dyadic(k);
            then consider i being Nat such that
            i <= (2|^k) and
A15:        o = i/(2|^k) by URYSOHN1:def 1;
A16:        0 < 2|^s0 by NEWTON:83;
            0 < 2|^k by NEWTON:83;
            then
A17:        1*(2|^k) = i*(2|^s0) by A15,A16,XCMPLX_1:95;
            then
A18:        i = (2|^k)/(2|^s0) by A16,XCMPLX_1:89;
A19:        not ex n being Nat st i = n + 1
            proof
              given n being Nat such that
A20:          i = n + 1;
              n + 1 -1 < 0 by A18,A14,A20,XREAL_1:49,83;
              hence thesis;
            end;
            i <> 0 by A17,NEWTON:83;
            hence thesis by A19,NAT_1:6;
          end;
          1 <= (2|^s0) by PREPOWER:11;
          then o in dyadic(s0) by URYSOHN1:def 1;
          hence thesis by A10,A13,FUNCT_2:def 1;
        end;
        suppose
A21:      s0 < k;
          set o = 1/(2|^k);
A22:      not o in dyadic(s0)
          proof
A23:        (2|^s0) < 1 * (2|^k) by A21,PEPIN:66;
            assume o in dyadic(s0);
            then consider i being Nat such that
            i <= (2|^s0) and
A24:        o = i/(2|^s0) by URYSOHN1:def 1;
A25:        0 < 2|^k by NEWTON:83;
            0 < 2|^s0 by NEWTON:83;
            then
A26:        1*(2|^s0) = i*(2|^k) by A24,A25,XCMPLX_1:95;
            then
A27:        i = (2|^s0)/(2|^k) by A25,XCMPLX_1:89;
A28:        not ex n being Nat st i = n + 1
            proof
              given n being Nat such that
A29:          i = n + 1;
              n + 1 -1 < 0 by A27,A23,A29,XREAL_1:49,83;
              hence thesis;
            end;
            i <> 0 by A26,NEWTON:83;
            hence thesis by A28,NAT_1:6;
          end;
          1 <= (2|^k) by PREPOWER:11;
          then o in dyadic(k) by URYSOHN1:def 1;
          hence thesis by A10,A22,FUNCT_2:def 1;
        end;
      end;
      hence thesis;
    end;
    reconsider xx as Element of PFuncs(DYADIC,bool the carrier of T) by Th3;
    reconsider xx as Element of D;
A30: yy is Element of PFuncs(DYADIC,bool the carrier of T) by Th3;
    then reconsider yy as Element of D by A1;
    ex y being Element of D st P1[x,y]
    proof
      take yy;
      reconsider yy as PartFunc of DYADIC, bool the carrier of T by A30,
PARTFUN1:46;
      reconsider xx as PartFunc of DYADIC, bool the carrier of T by PARTFUN1:47
;
      take xx,yy;
      thus thesis by A9,A7;
    end;
    then consider y being Element of D such that
A31: P1[x,y];
    take y;
    thus thesis by A31;
  end;
  ex F being sequence of D st F.0 = S & for n being Nat
  holds Q[n,F.n,F.(n+1)] from RECDEF_1:sch 2(A4);
  then consider F being sequence of D such that
A32: F.0 = S and
A33: for n being Nat holds P1[F.n,F.(n+1)];
  for x being object holds x in D implies x in PFuncs(DYADIC,bool the
  carrier of T) by A1;
  then rng F c= D & D c= PFuncs(DYADIC,bool the carrier of T) by RELAT_1:def 19
;
  then
A34: dom F = NAT & rng F c= PFuncs(DYADIC,bool the carrier of T) by
FUNCT_2:def 1;
  defpred R[Nat,PartFunc of DYADIC,bool the carrier of T,
PartFunc
  of DYADIC,bool the carrier of T] means ($2=F.$1 & $3 = F.($1+1) & (ex k being
  Nat st $2 is Drizzle of A,B,k) & (for k being Nat st $2
  is Drizzle of A,B,k holds $3 is Drizzle of A,B,k+1) & (for r being Element of
  dom $2 holds $2.r = $3.r));
  reconsider F as Functional_Sequence of DYADIC,bool the carrier of T by A34,
FUNCT_2:def 1,RELSET_1:4;
  defpred P[Nat] means F.$1 is Drizzle of A,B,$1 &
   for r being Element of dom (F.$1) holds (F.$1).r = (F.($1+1)).r;
A35: for n being Nat st P[n] holds P[n+1]
  proof
    let n be Nat;
    assume
A36: P[n];
    ex xx,yy being PartFunc of DYADIC,bool the carrier of T st R[n,xx,yy]
    by A33;
    hence F.(n+1) is Drizzle of A,B,(n+1) by A36;
    let r be Element of dom (F.(n+1));
    ex xx1,yy1 being PartFunc of DYADIC,bool the carrier of T st R[n+1,xx1
    ,yy1] by A33;
    hence thesis;
  end;
  take F;
  ex xx,yy being PartFunc of DYADIC,bool the carrier of T st R[0,xx,yy] by A33;
  then
A37: P[0] by A32;
  for n being Nat holds P[n] from NAT_1:sch 2(A37,A35);
  hence thesis;
end;
