reserve T,T1,T2 for TopSpace,
  A,B for Subset of T,
  F for Subset of T|A,
  G,G1, G2 for Subset-Family of T,
  U,W for open Subset of T|A,
  p for Point of T|A,
  n for Nat,
  I for Integer;

theorem Th1:
  F = B/\A implies Fr F c= Fr B/\A
proof
A1: [#](T|A)=A by PRE_TOPC:def 5;
  [#](T|A)c=[#]T by PRE_TOPC:def 4;
  then reconsider F9=F,Fd=F` as Subset of T by XBOOLE_1:1;
  assume
A2: F=B/\A;
  then Cl F9 c=Cl B by PRE_TOPC:19,XBOOLE_1:18;
  then Cl F9/\A c=Cl B/\A by XBOOLE_1:26;
  then
A3: Cl F c=Cl B/\A by A1,PRE_TOPC:17;
  [#](T|A)=A by PRE_TOPC:def 5;
  then F`=A\B by A2,XBOOLE_1:47;
  then F`c=B` by XBOOLE_1:35;
  then Cl Fd c=Cl B` by PRE_TOPC:19;
  then
A4: Cl Fd/\A c=Cl B`/\A by XBOOLE_1:26;
  Cl F`=Cl Fd/\[#](T|A) by PRE_TOPC:17;
  then Cl F`c=Cl B` by A1,A4,XBOOLE_1:18;
  then Cl F/\Cl F`c=(Cl B/\A)/\Cl B` by A3,XBOOLE_1:27;
  hence thesis by XBOOLE_1:16;
end;
