reserve x,y for Real,
  u,v,w for set,
  r for positive Real;

theorem Th52:
  for X being T_1 non empty TopSpace for B being prebasis of X st
  for x being Point of X for V being Subset of X st x in V & V in B ex f being
continuous Function of X, I[01] st f.x = 0 & f.:V` c= {1} holds X is Tychonoff
proof
  reconsider z = 0 as Point of I[01] by BORSUK_1:40,XXREAL_1:1;
  let X be T_1 non empty TopSpace;
  let BB be prebasis of X such that
A1: for x being Point of X for V being Subset of X st x in V & V in BB
  ex f being continuous Function of X, I[01] st f.x = 0 & f.:V` c= {1};
  let A be closed Subset of X;
  let a be Point of X;
A2: FinMeetCl BB is Basis of X by YELLOW_9:23;
  assume a in A`;
  then consider B being Subset of X such that
A3: B in FinMeetCl BB and
A4: a in B and
A5: B c= A` by A2,YELLOW_9:31;
  consider F being Subset-Family of X such that
A6: F c= BB and
A7: F is finite and
A8: B = Intersect F by A3,CANTOR_1:def 3;
  per cases;
  suppose
    F is empty;
    then B = the carrier of X by A8,SETFAM_1:def 9;
    then A`` = {}X by A5,XBOOLE_1:37;
    then
A9: (X-->z).:A = {};
    (X-->z).a = z;
    hence thesis by A9,XBOOLE_1:2;
  end;
  suppose
    F is non empty;
    then reconsider F as finite non empty Subset-Family of X by A7;
    defpred P[object,object] means
   ex S being Subset of X, f being continuous
    Function of X, I[01] st S = $1 & f = $2 & f.a = 0 & f.:S` c= {1};
    reconsider Sa = {In(0,REAL)}
           as finite non empty Subset of REAL;
    set z = the Element of F;
    set R = I[01];
A10: for x being object st x in F ex y being object st P[x,y]
    proof
      let x be object;
      assume
A11:  x in F;
      then reconsider S = x as Subset of X;
      a in S by A4,A8,A11,SETFAM_1:43;
      then consider f being continuous Function of X, I[01] such that
A12:  f.a = 0 and
A13:  f.:S` c= {1} by A6,A11,A1;
      take f;
      thus thesis by A12,A13;
    end;
    consider G being Function such that
A14: dom G = F & for x being object st x in F holds P[x,G.x] from
    CLASSES1:sch 1(A10);
    G is Function-yielding
    proof
      let x be object;
      assume x in dom G;
      then P[x,G.x] by A14;
      hence thesis;
    end;
    then reconsider G as ManySortedFunction of F by A14,PARTFUN1:def 2
,RELAT_1:def 18;
    rng G c= Funcs(the carrier of X,the carrier of R)
    proof
      let u be object;
      assume u in rng G;
      then consider v being object such that
A15:  v in dom G and
A16:  u = G.v by FUNCT_1:def 3;
      P[v,u] by A14,A15,A16;
      hence thesis by FUNCT_2:8;
    end;
    then G is Function of F, Funcs(the carrier of X,the carrier of R) by A14,
FUNCT_2:2;
    then
A17: G in Funcs(F, Funcs(the carrier of X,the carrier of R)) by FUNCT_2:8;
    then commute G in Funcs(the carrier of X, Funcs(F, the carrier of R)) by
FUNCT_6:55;
    then reconsider
    cG = commute G as Function of the carrier of X, Funcs(F, the
    carrier of R) by FUNCT_2:66;
    now
      let a be set;
      assume a in F;
      then P[a,G.a] by A14;
      hence G.a is continuous Function of X, I[01];
    end;
    then consider f being continuous Function of X, I[01] such that
A18: for x being Point of X, S being finite non empty Subset of REAL
    st S = rng ((commute G).x) holds f.x = max S by Th51;
    take f;
    reconsider cGa = cG.a as Function of F, the carrier of R;
A19: dom cGa = F by FUNCT_2:def 1;
    Sa = rng ((commute G).a)
    proof
      thus Sa c= rng ((commute G).a)
      proof
        let x be object;
        assume x in Sa;
        then
A20:    x = 0 by TARSKI:def 1;
        P[z,G.z] by A14;
        then x = (commute G).a.z by A20,A17,FUNCT_6:56;
        hence thesis by A19,FUNCT_1:def 3;
      end;
      let x be object;
      assume x in rng ((commute G).a);
      then consider z being object such that
A21:  z in dom cGa and
A22:  x = cGa.z by FUNCT_1:def 3;
      P[z,G.z] by A14,A21;
      then x = 0 by A17,A21,A22,FUNCT_6:56;
      hence thesis by TARSKI:def 1;
    end;
    hence f.a = max Sa by A18
      .= 0 by XXREAL_2:11;
    let z be object;
    assume z in f.:A;
    then consider x being object such that
A23: x in dom f and
A24: x in A and
A25: z = f.x by FUNCT_1:def 6;
    reconsider x as Element of X by A23;
    not x in B by A5,A24,XBOOLE_0:def 5;
    then consider w such that
A26: w in F and
A27: not x in w by A8,SETFAM_1:43;
    reconsider cGx = cG.x as Function of F, the carrier of R;
    reconsider S = rng cGx as finite non empty Subset of REAL by BORSUK_1:40
,XBOOLE_1:1;
A28: f.x = max S by A18;
    consider T being Subset of X, g being continuous Function of X, R such
    that
A29: T = w and
A30: g = G.w and
    g.a = 0 and
A31: g.:T` c= {1} by A14,A26;
    x in T` by A27,A29,SUBSET_1:29;
    then g.x in g.:T` by FUNCT_2:35;
    then g.x = 1 by A31,TARSKI:def 1;
    then
A32: cGx.w = 1 by A17,A26,A30,FUNCT_6:56;
    w in dom cGx by A17,A26,A30,FUNCT_6:56;
    then
A33: 1 in S by A32,FUNCT_1:def 3;
    for r being ExtReal st r in S holds r <= 1 by BORSUK_1:40,XXREAL_1:1;
    then max S = 1 by A33,XXREAL_2:def 8;
    hence thesis by A25,A28,TARSKI:def 1;
  end;
end;
