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 Th22:
  weight TM c= iC iff density TM c= iC
proof
  consider A be Subset of TM such that
A1: A is dense and
A2: density TM=card A by TOPGEN_1:def 12;
  hereby
    assume weight TM c=iC;
    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
Th21;
    hence density TM c=iC by Lm6;
  end;
A3: weight TM c=omega*`card A by A1,Th17;
  assume density TM c=iC;
  then omega*`card A c=omega*`iC by A2,CARD_2:90;
  then weight TM c=omega*`iC by A3;
  hence thesis by Lm5;
end;
