
theorem
  for T being TopSpace, F, G being Subset-Family of T holds Fr (F \/ G)
  = Fr F \/ Fr G
proof
  let T be TopSpace, F, G be Subset-Family of T;
  thus Fr (F \/ G) c= Fr F \/ Fr G
  proof
    let x be object;
    assume
A1: x in Fr (F \/ G);
    then reconsider A = x as Subset of T;
    consider B being Subset of T such that
A2: A = Fr B and
A3: B in F \/ G by A1,Def1;
    per cases by A3,XBOOLE_0:def 3;
    suppose
      B in F;
      then A in Fr F by A2,Def1;
      hence thesis by XBOOLE_0:def 3;
    end;
    suppose
      B in G;
      then A in Fr G by A2,Def1;
      hence thesis by XBOOLE_0:def 3;
    end;
  end;
  Fr F c= Fr (F \/ G) & Fr G c= Fr (F \/ G) by Th6,XBOOLE_1:7;
  hence thesis by XBOOLE_1:8;
end;
