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 Th14:
  for TM st for Am st Am is closed & Am is discrete holds card Am
  c= iC holds for Am st Am is discrete holds card Am c= iC
proof
  let TM;
  assume
A1: for Am st Am is closed & Am is discrete holds card Am c=iC;
  let Am such that
A2: Am is discrete;
  per cases;
  suppose
    Am is empty;
    hence thesis;
  end;
  suppose
A3: Am is non empty;
    then reconsider Tm=TM as metrizable non empty TopSpace;
    Am c=Cl Am by PRE_TOPC:18;
    then reconsider ClA=Cl Am as non empty closed Subset of Tm by A3;
    set TA=Tm|ClA;
    reconsider A9=Am as open Subset of TA by A2,Th12;
    consider F be closed countable Subset-Family of TA such that
A4: A9=union F by TOPGEN_4:def 6;
    consider f be sequence of F such that
A5: rng f=F by A3,A4,CARD_3:96,ZFMISC_1:2;
A6: for x be object st x in dom f holds card(f.x)c=iC
    proof
      let x be object;
      assume x in dom f;
      then
A7:   f.x in rng f by FUNCT_1:def 3;
      then reconsider fx=f.x as Subset of TA by A5;
A8:   f.x c=Am by A4,A7,ZFMISC_1:74;
      then reconsider Fx=f.x as Subset of TM by XBOOLE_1:1;
      [#]TA=ClA & fx is closed by A7,PRE_TOPC:def 5,TOPS_2:def 2;
      then Fx is closed by TSEP_1:8;
      hence thesis by A1,A2,A8,TEX_2:22;
    end;
    card dom f=omega by A3,A4,CARD_1:47,FUNCT_2:def 1,ZFMISC_1:2;
    then card Union f c=(omega)*`iC by A6,CARD_2:86;
    hence thesis by A4,A5,Lm5;
  end;
end;
