reserve T for 1-sorted;
reserve T for TopSpace;

theorem Th28:
  for A,B,C being Subset of T st A is closed_condensed & B is
closed_condensed & C is closed_condensed holds Cl(Int(A /\ (Cl(Int(B /\ C)))))
  = Cl(Int((Cl(Int(A /\ B)) /\ C)))
proof
  let A,B,C be Subset of T;
  assume that
A1: A is closed_condensed and
A2: B is closed_condensed and
A3: C is closed_condensed;
A4: B = Cl Int B by A2,TOPS_1:def 7;
A5: (A /\ B) /\ C c= C by XBOOLE_1:17;
A6: Int(A /\ Cl Int(B /\ C)) c= A /\ Cl Int(B /\ C) by TOPS_1:16;
A7: Cl Int(B /\ C) = Cl(Int B /\ Int C) by TOPS_1:17;
  C = Cl Int C by A3,TOPS_1:def 7;
  then A /\ Cl Int(B /\ C) c= A /\ (B /\ C) by A7,A4,PRE_TOPC:21,XBOOLE_1:26;
  then
A8: Int(A /\ Cl Int(B /\ C)) c= A /\ (B /\ C) by A6;
  then Int(A /\ Cl Int(B /\ C)) c= (A /\ B) /\ C by XBOOLE_1:16;
  then
A9: Int(A /\ Cl Int(B /\ C)) c= C by A5;
A10: (A /\ B) /\ C c= A /\ B by XBOOLE_1:17;
  Int(A /\ Cl Int(B /\ C)) c= (A /\ B) /\ C by A8,XBOOLE_1:16;
  then Int(A /\ Cl Int(B /\ C)) c= A /\ B by A10;
  then
A11: Int Int(A /\ Cl Int(B /\ C)) c= Int(A /\ B) by TOPS_1:19;
  Int(A /\ B) c= Cl Int(A /\ B) by PRE_TOPC:18;
  then Int(A /\ Cl Int(B /\ C)) c= Cl Int(A /\ B) by A11;
  then Int(A /\ Cl Int(B /\ C)) c= Cl Int(A /\ B) /\ C by A9,XBOOLE_1:19;
  then Int Int(A /\ Cl Int(B /\ C)) c= Int(Cl Int(A /\ B) /\ C) by TOPS_1:19;
  then
A12: Cl Int(A /\ Cl Int(B /\ C)) c= Cl Int(Cl Int(A /\ B) /\ C) by PRE_TOPC:19;
A13: A /\ (B /\ C) c= B /\ C by XBOOLE_1:17;
A14: A /\ (B /\ C) c= A by XBOOLE_1:17;
A15: Int(Cl Int(A /\ B) /\ C) c= Cl Int(A /\ B) /\ C by TOPS_1:16;
A16: Cl Int(A /\ B) = Cl(Int A /\ Int B) by TOPS_1:17;
  A = Cl Int A by A1,TOPS_1:def 7;
  then Cl Int(A /\ B) /\ C c= (A /\ B) /\ C by A16,A4,PRE_TOPC:21,XBOOLE_1:26;
  then
A17: Int(Cl Int(A /\ B) /\ C) c= (A /\ B) /\ C by A15;
  then Int(Cl Int(A /\ B) /\ C) c= A /\ (B /\ C) by XBOOLE_1:16;
  then
A18: Int(Cl Int(A /\ B) /\ C) c= A by A14;
  Int(Cl Int(A /\ B) /\ C) c= A /\ (B /\ C) by A17,XBOOLE_1:16;
  then Int(Cl Int(A /\ B) /\ C) c= B /\ C by A13;
  then
A19: Int Int(Cl Int(A /\ B) /\ C) c= Int(B /\ C) by TOPS_1:19;
  Int(B /\ C) c= Cl Int(B /\ C) by PRE_TOPC:18;
  then Int(Cl Int(A /\ B) /\ C) c= Cl Int(B /\ C) by A19;
  then Int(Cl Int(A /\ B) /\ C) c= A /\ Cl Int(B /\ C) by A18,XBOOLE_1:19;
  then Int Int(Cl Int(A /\ B) /\ C) c= Int(A /\ Cl Int(B /\ C)) by TOPS_1:19;
  then Cl Int(Cl Int(A /\ B) /\ C) c= Cl Int(A /\ Cl Int(B /\ C)) by
PRE_TOPC:19;
  hence thesis by A12;
end;
