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 Th6:
  T1 is non empty & T2 is non empty implies weight T1 c= weight [:
  T1,T2:] & weight T2 c= weight[:T1,T2:]
proof
  defpred P[object] means not contradiction;
  set PR2=pr2(the carrier of T1,the carrier of T2);
  set PR1=pr1(the carrier of T1,the carrier of T2);
  assume T1 is non empty & T2 is non empty;
  then reconsider t1=T1,t2=T2 as non empty TopSpace;
  reconsider T12=[:t1,t2:] as non empty TopSpace;
  consider B12 be Basis of T12 such that
A1: card B12=weight T12 by WAYBEL23:74;
  deffunc F1(Subset of T12)=PR1.:$1;
  defpred PP[object] means $1 in B12 & P[$1];
  consider B1 be Subset-Family of T1 such that
A2: B1={F1(w) where w is Subset of T12:PP[w]} from LMOD_7:sch 5;
A3: B12 c=the topology of T12 by TOPS_2:64;
A4: B1 c=the topology of T1
  proof
    let x be object;
    assume x in B1;
    then consider w be Subset of T12 such that
A5: x=F1(w) and
A6: PP[w] by A2;
    w is open by A3,A6;
    then F1(w) is open by BORSUK_1:18;
    hence thesis by A5;
  end;
  for A be Subset of T1 st A is open for p be Point of T1 st p in A ex a
  be Subset of T1 st a in B1 & p in a & a c=A
  proof
    let A be Subset of T1;
    assume A is open;
    then
A7: [:A,[#]T2:] is open by BORSUK_1:6;
    set p2=the Point of T2;
A8: p2 in [#]t2;
    let p1 be Point of T1;
    assume p1 in A;
    then
A9: [p1,p2] in [:A,[#]T2:] by A8,ZFMISC_1:87;
    then reconsider p=[p1,p2] as Point of T12;
    consider a12 be Subset of T12 such that
A10: a12 in B12 and
A11: p in a12 and
A12: a12 c=[:A,[#]T2:] by A7,A9,YELLOW_9:31;
    p1 in [#]t1 & p2 in [#]t2;
    then
A13: PR1.(p1,p2)=p1 by FUNCT_3:def 4;
    a12 is open by A3,A10;
    then reconsider a=F1(a12) as open Subset of T1 by BORSUK_1:18;
    take a;
    dom PR1 = [:[#]T1,[#]T2:] by FUNCT_3:def 4
      .= [#]T12 by BORSUK_1:def 2;
    hence a in B1 & p1 in a by A2,A10,A11,A13,FUNCT_1:def 6;
    let y be object;
    assume y in a;
    then consider x be object such that
A14: x in dom PR1 and
A15: x in a12 & y=PR1.x by FUNCT_1:def 6;
    consider x1,x2 be object such that
A16: x1 in [#]T1 & x2 in [#]T2 and
A17: x=[x1,x2] by A14,ZFMISC_1:def 2;
    PR1.(x1,x2)=x1 by A16,FUNCT_3:def 4;
    hence thesis by A12,A15,A17,ZFMISC_1:87;
  end;
  then reconsider B1 as Basis of T1 by A4,YELLOW_9:32;
A18: card{F1(w) where w is Subset of T12:PP[w]} c= card B12 from BORSUK_2:
  sch 1;
  weight t1 c=card B1 by WAYBEL23:73;
  hence weight T1 c=weight[:T1,T2:] by A1,A2,A18;
  deffunc F2(Subset of T12)=PR2.:$1;
  consider B2 be Subset-Family of T2 such that
A19: B2={F2(w) where w is Subset of T12:PP[w]} from LMOD_7:sch 5;
A20: for A be Subset of T2 st A is open for p be Point of T2 st p in A ex a
  be Subset of T2 st a in B2 & p in a & a c=A
  proof
    let A be Subset of T2;
    assume A is open;
    then
A21: [:[#]T1,A:] is open by BORSUK_1:6;
    set p1=the Point of T1;
A22: p1 in [#]t1;
    let p2 be Point of T2;
    assume p2 in A;
    then
A23: [p1,p2] in [:[#]T1,A:] by A22,ZFMISC_1:87;
    then reconsider p=[p1,p2] as Point of T12;
    consider a12 be Subset of T12 such that
A24: a12 in B12 and
A25: p in a12 and
A26: a12 c=[:[#]T1,A:] by A21,A23,YELLOW_9:31;
    p1 in [#]t1 & p2 in [#]t2;
    then
A27: PR2.(p1,p2)=p2 by FUNCT_3:def 5;
    a12 is open by A3,A24;
    then reconsider a=F2(a12) as open Subset of T2 by BORSUK_1:18;
    take a;
    dom PR2 = [:[#]T1,[#]T2:] by FUNCT_3:def 5
      .= [#]T12 by BORSUK_1:def 2;
    hence a in B2 & p2 in a by A19,A24,A25,A27,FUNCT_1:def 6;
    let y be object;
    assume y in a;
    then consider x be object such that
A28: x in dom PR2 and
A29: x in a12 & y=PR2.x by FUNCT_1:def 6;
    consider x1,x2 be object such that
A30: x1 in [#]T1 & x2 in [#]T2 and
A31: x=[x1,x2] by A28,ZFMISC_1:def 2;
    PR2.(x1,x2)=x2 by A30,FUNCT_3:def 5;
    hence thesis by A26,A29,A31,ZFMISC_1:87;
  end;
  B2 c=the topology of T2
  proof
    let x be object;
    assume x in B2;
    then consider w be Subset of T12 such that
A32: x=F2(w) and
A33: PP[w] by A19;
    w is open by A3,A33;
    then F2(w) is open by BORSUK_1:18;
    hence thesis by A32;
  end;
  then reconsider B2 as Basis of T2 by A20,YELLOW_9:32;
A34: card{F2(w) where w is Subset of T12:PP[w]}c=card B12 from BORSUK_2:sch
  1;
  weight T2 c=card B2 by WAYBEL23:73;
  hence thesis by A1,A19,A34;
end;
