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 Th7:
  card (F|A) c= card F
proof
  set FA=F|A;
  per cases;
  suppose
    FA is empty;
    hence thesis;
  end;
  suppose
A1: FA is non empty;
    deffunc F(set) = $1/\A;
A2: A=[#](T|A) by PRE_TOPC:def 5;
A3: for x be set st x in F holds F(x) in FA
    proof
      let x be set;
A4:   F(x)c=A by XBOOLE_1:17;
      assume x in F;
      hence thesis by A2,A4,TOPS_2:def 3;
    end;
    consider g be Function of F,FA such that
A5: for x be set st x in F holds g.x=F(x) from FUNCT_2:sch 11(A3);
A6: dom g=F by A1,FUNCT_2:def 1;
    FA c=rng g
    proof
      let x be object;
      assume x in FA;
      then consider B such that
A7:   B in F and
A8:   F(B)=x by TOPS_2:def 3;
      x=g.B by A5,A7,A8;
      hence thesis by A6,A7,FUNCT_1:def 3;
    end;
    hence thesis by A6,CARD_1:12;
  end;
end;
