 reserve n,m for Nat,
         p,q for Point of TOP-REAL n, r for Real;
reserve M,M1,M2 for non empty TopSpace;

theorem
  for M be compact locally_euclidean non empty TopSpace
    ex P be a_partition of the carrier of M st
      for A be Subset of M st A in P holds
        A is open a_component &
        ex n st M|A is n-locally_euclidean non empty TopSpace
  proof
    let M be compact locally_euclidean non empty TopSpace;
    set P={Component_of p where p is Point of M: not contradiction};
    P c= bool the carrier of M
    proof
      let x be object;
      assume x in P;
      then ex p be Point of M st x=Component_of p & not contradiction;
      hence thesis;
    end;
    then reconsider P as Subset-Family of M;
A1: the carrier of M c= union P
    proof
      let x be object;
      assume x in the carrier of M;
      then reconsider x as Point of M;
A2:   Component_of x in P;
      x in Component_of x by CONNSP_1:38;
      hence thesis by A2,TARSKI:def 4;
    end;
    for A be Subset of M st A in P holds A<>{} & for B be Subset of M st
      B in P holds A = B or A misses B
    proof
      let A be Subset of M;
      assume A in P;
      then consider p be Point of M such that
A3:   A=Component_of p
      and not contradiction;
      thus A <>{} by A3;
      let B be Subset of M such that
A4:     B in P
      and
A5:     B<>A;
      A6:ex q be Point of M st B=Component_of q & not contradiction by A4;
      assume A meets B;
      then consider x be object such that
A7:   x in A
      and
A8:   x in B by XBOOLE_0:3;
      reconsider x as Point of M by A7;
      Component_of p = Component_of x by CONNSP_1:42,A7,A3;
      hence thesis by CONNSP_1:42,A8,A6,A5,A3;
    end;
    then reconsider P as a_partition of the carrier of M
      by A1,XBOOLE_0:def 10,EQREL_1:def 4;
    take P;
    let A be Subset of M;
    assume A in P;
    then ex p be Point of M st A=Component_of p & not contradiction;
    then A is a_component by CONNSP_1:40;
    hence thesis by Th14;
  end;
