reserve T for TopSpace;

theorem Th21:
  for F,G being Subset-Family of T holds Int(F \/ G) = (Int F) \/ (Int G)
proof
  let F,G be Subset-Family of T;
  for X being object holds X in Int(F \/ G) iff X in (Int F) \/ (Int G)
  proof
    let X be object;
A1: now
      assume
A2:   X in (Int F) \/ (Int G);
      now
        per cases by A2,XBOOLE_0:def 3;
        suppose
A3:       X in Int F;
          then reconsider X0 = X as Subset of T;
          ex W being Subset of T st X0 = Int W & W in (F \/ G)
          proof
            consider Z being Subset of T such that
A4:         X0 = Int Z and
A5:         Z in F by A3,Def1;
            take Z;
            thus thesis by A4,A5,XBOOLE_0:def 3;
          end;
          hence X in Int(F \/ G) by Def1;
        end;
        suppose
A6:       X in Int G;
          then reconsider X0 = X as Subset of T;
          ex W being Subset of T st X0 = Int W & W in (F \/ G)
          proof
            consider Z being Subset of T such that
A7:         X0 = Int Z and
A8:         Z in G by A6,Def1;
            take Z;
            thus thesis by A7,A8,XBOOLE_0:def 3;
          end;
          hence X in Int(F \/ G) by Def1;
        end;
      end;
      hence X in Int(F \/ G);
    end;
    now
      assume
A9:   X in Int(F \/ G);
      then reconsider X0 = X as Subset of T;
      consider W being Subset of T such that
A10:  X0 = Int W and
A11:  W in (F \/ G) by A9,Def1;
      now
        per cases by A11,XBOOLE_0:def 3;
        suppose
          W in F;
          then X0 in Int F by A10,Def1;
          hence X0 in (Int F) \/ (Int G) by XBOOLE_0:def 3;
        end;
        suppose
          W in G;
          then X0 in Int G by A10,Def1;
          hence X0 in (Int F) \/ (Int G) by XBOOLE_0:def 3;
        end;
      end;
      hence X in (Int F) \/ (Int G);
    end;
    hence thesis by A1;
  end;
  hence thesis by TARSKI:2;
end;
