reserve x, y, z for set,
  T for TopStruct,
  A for SubSpace of T,
  P, Q for Subset of T;
reserve TS for TopSpace;
reserve PS, QS for Subset of TS;

theorem
  for TS being non empty TopSpace holds TS is T_2 & TS is compact
  implies TS is normal
proof
  let TS be non empty TopSpace;
  assume that
A1: TS is T_2 and
A2: TS is compact;
  let A,B be Subset of TS such that
A3: A <> {} and
A4: B <> {} and
A5: A is closed and
A6: B is closed and
A7: A /\ B = {};
  defpred P[object,object,object] means
ex P,Q being Subset of TS st $2 = P & $3 = Q &
  P is open & Q is open & $1 in P & B c= Q & P /\ Q = {};
A8: for x being object holds x in A implies
   ex y,z being object st y in the topology of TS & z in the topology of
  TS & P[x,y,z]
  proof let x be object;
    assume
A9: x in A;
    then reconsider p=x as Point of TS;
    not p in B by A7,A9,XBOOLE_0:def 4;
    then p in B` by SUBSET_1:29;
    then consider W,V being Subset of TS such that
A10: W is open and
A11: V is open and
A12: p in W and
A13: B c= V and
A14: W misses V by A1,A2,A4,A6,Th6,Th8;
    reconsider X = W, Y = V as set;
    take X,Y;
    thus X in the topology of TS & Y in the topology of TS by A10,A11;
    W /\ V = {} by A14;
    hence thesis by A10,A11,A12,A13;
  end;
  consider f,g being Function such that
A15: dom f = A & dom g = A and
A16: for x being object st x in A holds P[x,f.x,g.x] from MCART_1:sch 1(A8);
  f.:A c= bool the carrier of TS
  proof
    let x be object;
    assume x in f.:A;
    then consider y being object such that
    y in dom f and
A17: y in A and
A18: f.y=x by FUNCT_1:def 6;
    ex P,Q being Subset of TS st f.y=P & g.y=Q & P is open & Q is open &
    y in P & B c= Q & P /\ Q = {} by A16,A17;
    hence thesis by A18;
  end;
  then reconsider Cf = f.:A as Subset-Family of TS;
  A c= union Cf
  proof
    let x be object;
    assume
A19: x in A;
    then consider P,Q being Subset of TS such that
A20: f.x=P and
    g.x=Q and
    P is open and
    Q is open and
A21: x in P and
    B c= Q and
    P /\ Q = {} by A16;
    P in Cf by A15,A19,A20,FUNCT_1:def 6;
    hence thesis by A21,TARSKI:def 4;
  end;
  then
A22: Cf is Cover of A by SETFAM_1:def 11;
A23: Cf is open
  proof
    let G be Subset of TS;
    assume G in Cf;
    then consider x being object such that
    x in dom f and
A24: x in A and
A25: G = f.x by FUNCT_1:def 6;
    ex P,Q being Subset of TS st f.x=P & g.x=Q & P is open & Q is open &
    x in P & B c= Q & P /\ Q = {} by A16,A24;
    hence thesis by A25;
  end;
  A is compact by A2,A5,Th8;
  then consider C being Subset-Family of TS such that
A26: C c= Cf and
A27: C is Cover of A and
A28: C is finite by A22,A23;
  set z = the Element of A;
  A c= union C by A27,SETFAM_1:def 11;
  then z in union C by A3;
  then consider D being set such that
  z in D and
A29: D in C by TARSKI:def 4;
  C c= rng f by A15,A26,RELAT_1:113;
  then consider H9 being set such that
A30: H9 c= dom f and
A31: H9 is finite and
A32: f.:H9 = C by A28,ORDERS_1:85;
  g.:H9 c= bool the carrier of TS
  proof
    let x be object;
    assume x in g.:H9;
    then consider y being object such that
    y in dom g and
A33: y in H9 and
A34: g.y=x by FUNCT_1:def 6;
    ex P,Q being Subset of TS st f.y=P & g.y=Q & P is open & Q is open &
    y in P & B c= Q & P /\ Q = {} by A15,A16,A30,A33;
    hence thesis by A34;
  end;
  then reconsider Bk = g.:H9 as Subset-Family of TS;
  consider y being object such that
A35: y in dom f and
A36: y in H9 and
  D = f.y by A32,A29,FUNCT_1:def 6;
A37: for X being set st X in Bk holds B c= X
  proof
    let X be set;
    assume
A38: X in Bk;
    then reconsider X9 = X as Subset of TS;
    consider x being object such that
A39: x in dom g and
    x in H9 and
A40: X9 = g.x by A38,FUNCT_1:def 6;
    ex P,Q being Subset of TS st f.x = P & g.x = Q & P is open & Q is
    open & x in P & B c= Q & P /\ Q = {} by A15,A16,A39;
    hence thesis by A40;
  end;
  take W = union C, V = meet Bk;
  thus W is open by A23,A26,TOPS_2:11,19;
  Bk is open
  proof
    let G be Subset of TS;
    assume G in Bk;
    then consider x being object such that
A41: x in dom g and
    x in H9 and
A42: G = g.x by FUNCT_1:def 6;
    ex P,Q being Subset of TS st f.x = P & g.x = Q & P is open & Q is
    open & x in P & B c= Q & P /\ Q = {} by A15,A16,A41;
    hence thesis by A42;
  end;
  hence V is open by A31,TOPS_2:20;
  thus A c= W by A27,SETFAM_1:def 11;
  ex P,Q being Subset of TS st f.y = P & g.y = Q & P is open & Q is
  open & y in P & B c= Q & P /\ Q = {} by A15,A16,A35;
  then Bk <> {} by A15,A35,A36,FUNCT_1:def 6;
  hence B c= V by A37,SETFAM_1:5;
  thus W /\ V = {}
  proof
    set x = the Element of (union C) /\ (meet Bk);
    assume
A43: W /\ V <> {};
    then
A44: x in meet Bk by XBOOLE_0:def 4;
    x in (union C) by A43,XBOOLE_0:def 4;
    then consider D being set such that
A45: x in D and
A46: D in C by TARSKI:def 4;
    consider z being object such that
A47: z in dom f and
A48: z in H9 and
A49: D = f.z by A32,A46,FUNCT_1:def 6;
    consider P,Q being Subset of TS such that
A50: f.z = P and
A51: g.z = Q and
    P is open and
    Q is open and
    z in P and
    B c= Q and
A52: P /\ Q = {} by A15,A16,A47;
    Q in Bk by A15,A47,A48,A51,FUNCT_1:def 6;
    then x in Q by A44,SETFAM_1:def 1;
    hence contradiction by A45,A49,A50,A52,XBOOLE_0:def 4;
  end;
end;
