reserve n for Nat,
        X for set,
        Fx,Gx for Subset-Family of X;
reserve TM for metrizable TopSpace,
        TM1,TM2 for finite-ind second-countable metrizable TopSpace,
        A,B,L,H for Subset of TM,
        U,W for open Subset of TM,
        p for Point of TM,

        F,G for finite Subset-Family of TM,
        I for Integer;

theorem Th10:
  for T1,T2 be TopSpace,A1 be Subset of T1,A2 be Subset of T2 holds
  Fr [:A1,A2:] = [:Fr A1,Cl A2:] \/ [:Cl A1,Fr A2:]
proof
  let T1,T2 be TopSpace,A1 be Subset of T1,A2 be Subset of T2;
A1: [:Cl A1,Cl A2:]/\[:Cl A1`,[#]T2:]=[:(Cl A1)/\(Cl A1`),(Cl A2)/\[#]T2:] by
ZFMISC_1:100
    .=[:Fr A1,Cl A2:] by XBOOLE_1:28;
A2: [:Cl A1,Cl A2:]/\[:[#]T1,Cl A2`:]=[:(Cl A1)/\[#]T1,(Cl A2)/\(Cl A2`):] by
ZFMISC_1:100
    .=[:Cl A1,Fr A2:] by XBOOLE_1:28;
  Cl[#]T1=[#]T1 by TOPS_1:2;
  then
A3: Cl[:[#]T1,A2`:]=[:[#]T1,Cl A2`:] by TOPALG_3:14;
  Cl[#]T2=[#]T2 by TOPS_1:2;
  then
A4: Cl[:A1`,[#]T2:]=[:Cl A1`,[#]T2:] by TOPALG_3:14;
  Cl[:A1,A2:]`=Cl([:[#]T1,[#]T2:]\[:A1,A2:]) by BORSUK_1:def 2
    .=Cl([:[#]T1\A1,[#]T2:]\/[:[#]T1,[#]T2\A2:]) by ZFMISC_1:103
    .=[:Cl A1`,[#]T2:]\/[:[#]T1,Cl A2`:] by A4,A3,PRE_TOPC:20;
  hence Fr[:A1,A2:]=[:Cl A1,Cl A2:]/\([:Cl A1`,[#]T2:]\/[:[#]T1,Cl A2`:])
  by TOPALG_3:14
    .=[:Fr A1,Cl A2:]\/[:Cl A1,Fr A2:] by A1,A2,XBOOLE_1:23;
end;
