reserve T, T1, T2 for TopSpace,
  A, B for Subset of T,
  F, G for Subset-Family of T,
  A1 for Subset of T1,
  A2 for Subset of T2,
  TM, TM1, TM2 for metrizable TopSpace,
  Am, Bm for Subset of TM,
  Fm, Gm for Subset-Family of TM,
  C for Cardinal,
  iC for infinite Cardinal;

theorem Th18:
  weight TM c= iC iff for Fm st Fm is open & Fm is Cover of TM ex
  Gm st Gm c=Fm & Gm is Cover of TM & card Gm c= iC
proof
  hereby
    assume
A1: weight TM c=iC;
    let F be Subset-Family of TM such that
A2: F is open and
A3: F is Cover of TM;
    per cases;
    suppose
A4:   TM is empty;
      reconsider G={} as Subset-Family of TM by TOPGEN_4:18;
      take G;
      the carrier of TM = {} by A4;
      then [#]TM=union G;
      hence G c=F & G is Cover of TM & card G c=iC by SETFAM_1:def 11;
    end;
    suppose
      TM is non empty;
      then consider G be open Subset-Family of TM such that
A5:   G c=F & union G=union F & card G c=weight TM by A2,TOPGEN_2:11;
      reconsider G as Subset-Family of TM;
      take G;
      union F=[#]TM by A3,SETFAM_1:45;
      hence G c=F & G is Cover of TM & card G c=iC by A1,A5,SETFAM_1:def 11;
    end;
  end;
  assume for F be Subset-Family of TM st F is open & F is Cover of TM ex G
  be Subset-Family of TM st G c=F & G is Cover of TM & card G c=iC;
  then for A be Subset of TM st A is closed & A is discrete holds card A c=iC
  by Th13;
  then for A be Subset of TM st A is discrete holds card A c=iC by Th14;
  then for F be Subset-Family of TM st F is open & not{} in F & for A,B be
  Subset of TM st A in F & B in F & A<>B holds A misses B holds card F c=iC by
Th15;
  then density TM c=iC by Lm6;
  hence thesis by Lm7;
end;
