reserve PM for MetrStruct;
reserve x,y for Element of PM;
reserve r,p,q,s,t for Real;
reserve T for TopSpace;
reserve A for Subset of T;
reserve T for non empty TopSpace;
reserve x for Point of T;
reserve Z,X,V,W,Y,Q for Subset of T;
reserve FX for Subset-Family of T;
reserve a for set;
reserve x,y for Point of T;
reserve A,B for Subset of T;
reserve FX,GX for Subset-Family of T;

theorem
  T is T_2 & T is paracompact implies T is normal
proof
  assume that
A1: T is T_2 and
A2: T is paracompact;
  for A,B being Subset of T st A <> {} & B <> {} & A is closed & B is
closed & A misses B ex W,V being Subset of T st W is open & V is open & A c= W
  & B c= V & W misses V
  proof
    let A,B be Subset of T;
    assume that
A3: A <> {} and
    B <> {} and
A4: A is closed and
A5: B is closed and
A6: A misses B;
    for x st x in B ex W,V being Subset of T st W is open & V is open & A
    c= W & x in V & W misses V
    proof
      let x;
      assume x in B;
      then not x in A by A6,XBOOLE_0:3;
      then
A7:   x in A` by SUBSET_1:29;
      T is regular by A1,A2,Th24;
      then consider V,W being Subset of T such that
A8:   V is open & W is open & x in V & A c= W & V misses W by A3,A4,A7;
      take W,V;
      thus thesis by A8;
    end;
    then consider Y,Z being Subset of T such that
A9: Y is open & Z is open & A c= Y & B c= Z & Y misses Z by A2,A5,Th23;
    take Y,Z;
    thus thesis by A9;
  end;
  hence thesis;
end;
