reserve TS for 1-sorted,
  K, Q for Subset of TS;
reserve TS for TopSpace,
  GX for TopStruct,
  x for set,
  P, Q for Subset of TS,
  K , L for Subset of TS,
  R, S for Subset of GX,
  T, W for Subset of GX;

theorem Th68:
  P is closed_condensed & Q is closed_condensed implies P \/ Q is
  closed_condensed
proof
  assume that
A1: P is closed_condensed and
A2: Q is closed_condensed;
A3: Q = Cl(Int Q) by A2;
  P = Cl(Int P) by A1;
  then P \/ Q = Cl(Int P \/ Int Q) by A3,PRE_TOPC:20;
  then
A4: P \/ Q c= Cl(Int(P \/ Q)) by Th20,PRE_TOPC:19;
A5: Cl(Int(P \/ Q)) c= Cl(P \/ Q) by Th16,PRE_TOPC:19;
A6: Q is closed by A2;
  P is closed by A1;
  then Cl(Int(P \/ Q)) c= P \/ Q by A5,A6,PRE_TOPC:22;
  then P \/ Q = Cl(Int(P \/ Q)) by A4;
  hence thesis;
end;
