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

theorem Th17:
  for A,B being Subset of T st A is condensed & B is condensed
holds Int(Cl(A \/ B)) \/ (A \/ B) is condensed & Cl(Int(A /\ B)) /\ (A /\ B) is
  condensed
proof
  let A,B be Subset of T;
  assume
A1: A is condensed;
  assume
A2: B is condensed;
  then
A3: B c= Cl(Int(B)) by TOPS_1:def 6;
A4: A c= Cl(Int(A)) by A1,TOPS_1:def 6;
  thus Int(Cl(A \/ B)) \/ (A \/ B) is condensed
  proof
    set X = Int(Cl(A \/ B)) \/ (A \/ B);
    Cl(Int(A) \/ Int(B)) c= Cl(Int(A \/ B)) by PRE_TOPC:19,TOPS_1:20;
    then
A5: Cl(Int(A)) \/ Cl(Int(B)) c= Cl(Int(A \/ B)) by PRE_TOPC:20;
    A \/ B c= Cl(Int(A)) \/ Cl(Int(B)) by A4,A3,XBOOLE_1:13;
    then A \/ B c= Cl(Int(A \/ B)) by A5;
    then
A6: Int(Cl(A \/ B)) \/ (A \/ B) c= Int(Cl(A \/ B)) \/ Cl(Int(A \/ B)) by
XBOOLE_1:9;
    Cl(Int(Int(Cl(A \/ B))) \/ Int(A \/ B)) c= Cl(Int(X)) by PRE_TOPC:19
,TOPS_1:20;
    then
A7: Cl(Int(Cl(A \/ B))) \/ Cl(Int(A \/ B)) c= Cl(Int(X)) by PRE_TOPC:20;
    Cl(Int(Cl(A \/ B))) c= Cl(Cl(A \/ B)) by PRE_TOPC:19,TOPS_1:16;
    then Cl(Int(Cl(A \/ B))) \/ Cl(A \/ B) = Cl(A \/ B) by XBOOLE_1:12;
    then Int(Cl(X)) = Int(Cl(A \/ B)) by PRE_TOPC:20;
    then
A8: Int(Cl(X)) c= X by XBOOLE_1:7;
    Int(Cl(A \/ B)) \/ Cl(Int(A \/ B)) c= Cl(Int(Cl(A \/ B))) \/ Cl(Int(A
    \/ B)) by PRE_TOPC:18,XBOOLE_1:9;
    then Int(Cl(A \/ B)) \/ Cl(Int(A \/ B)) c= Cl(Int(X)) by A7;
    then X c= Cl(Int(X)) by A6;
    hence thesis by A8,TOPS_1:def 6;
  end;
A9: Int(Cl(B)) c= B by A2,TOPS_1:def 6;
A10: Int(Cl(A)) c= A by A1,TOPS_1:def 6;
  thus Cl(Int(A /\ B)) /\ (A /\ B) is condensed
  proof
    set X = Cl(Int(A /\ B)) /\ (A /\ B);
    Int(Cl(A /\ B)) c= Int(Cl(A) /\ Cl(B)) by PRE_TOPC:21,TOPS_1:19;
    then
A11: Int(Cl(A /\ B)) c= Int(Cl(A)) /\ Int(Cl(B)) by TOPS_1:17;
    Int(Cl(A)) /\ Int(Cl(B)) c= A /\ B by A10,A9,XBOOLE_1:27;
    then Int(Cl(A /\ B)) c= A /\ B by A11;
    then
A12: Cl(Int(A /\ B)) /\ Int(Cl(A /\ B)) c= Cl(Int(A /\ B)) /\ (A /\ B) by
XBOOLE_1:26;
    Int(Cl(X)) c= Int(Cl(Cl(Int(A /\ B))) /\ Cl(A /\ B)) by PRE_TOPC:21
,TOPS_1:19;
    then
A13: Int(Cl(X)) c= Int(Cl(Int(A /\ B))) /\ Int(Cl(A /\ B)) by TOPS_1:17;
    Int(Int(A /\ B)) c= Int(Cl(Int(A /\ B))) by PRE_TOPC:18,TOPS_1:19;
    then Int(A /\ B) = Int(Cl(Int(A /\ B))) /\ Int(A /\ B) by XBOOLE_1:28;
    then Cl(Int(A /\ B)) = Cl(Int(X)) by TOPS_1:17;
    then
A14: X c= Cl(Int(X)) by XBOOLE_1:17;
    Int(Cl(Int(A /\ B))) /\ Int(Cl(A /\ B)) c= Cl(Int(A /\ B)) /\ Int(Cl(
    A /\ B)) by TOPS_1:16,XBOOLE_1:26;
    then Int(Cl(X)) c= Cl(Int(A /\ B)) /\ Int(Cl(A /\ B)) by A13;
    then Int(Cl(X)) c= X by A12;
    hence thesis by A14,TOPS_1:def 6;
  end;
end;
