reserve T for non empty TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T;

theorem :: Ex. 1.3.B. a)
  F is locally_finite implies Fr union F c= union Fr F
proof
  assume F is locally_finite;
  then
A1: Cl union F = union clf F by PCOMPS_1:20;
  let x be object;
  assume x in Fr union F;
  then
A2: x in Cl union F /\ Cl (union F)` by TOPS_1:def 2;
  then
A3: x in Cl (union F)` by XBOOLE_0:def 4;
  x in Cl union F by A2,XBOOLE_0:def 4;
  then consider A being set such that
A4: x in A and
A5: A in clf F by A1,TARSKI:def 4;
  consider B being Subset of T such that
A6: A = Cl B and
A7: B in F by A5,PCOMPS_1:def 2;
  B c= union F by A7,ZFMISC_1:74;
  then (union F)` c= B` by SUBSET_1:12;
  then Cl (union F)` c= Cl B` by PRE_TOPC:19;
  then x in Cl B /\ Cl B` by A4,A6,A3,XBOOLE_0:def 4;
  then
A8: x in Fr B by TOPS_1:def 2;
  Fr B in Fr F by A7,TOPGEN_1:def 1;
  hence thesis by A8,TARSKI:def 4;
end;
