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 Th24:
  T is T_2 & T is paracompact implies T is regular
proof
  assume
A1: T is T_2;
  assume
A2: T is paracompact;
  for x for A being Subset of T st A <> {} & A is closed & x in A` ex W,V
  being Subset of T st W is open & V is open & x in W & A c= V & W misses V
  proof
    let x;
    let A be Subset of T;
    assume that
    A <> {} and
A3: A is closed and
A4: x in A`;
    set B = { x };
A5: not x in A by A4,XBOOLE_0:def 5;
    for y st y in A ex V,W being Subset of T st V is open & W is open & B
    c= V & y in W & V misses W
    proof
      let y;
      assume y in A;
      then consider V,W being Subset of T such that
A6:   V is open & W is open & x in V & y in W & V misses W by A1,A5,
PRE_TOPC:def 10;
      take V,W;
      thus thesis by A6,ZFMISC_1:31;
    end;
    then consider Y,Z being Subset of T such that
A7: Y is open & Z is open & B c= Y & A c= Z & Y misses Z by A2,A3,Th23;
    take Y,Z;
    thus thesis by A7,ZFMISC_1:31;
  end;
  hence thesis;
end;
