
theorem Th39:
  for V being RealUnitarySpace, M,N being Subset of V st M in
Family_open_set(V) & N in Family_open_set(V) holds M /\ N in Family_open_set(V)
proof
  let V be RealUnitarySpace;
  let M,N be Subset of V;
  assume that
A1: M in Family_open_set(V) and
A2: N in Family_open_set(V);
  for v being Point of V st v in M /\ N
   ex q being Real st q>0 & Ball(v,q)
  c= M /\ N
  proof
    let v be Point of V;
    assume
A3: v in M /\ N;
    then v in M by XBOOLE_0:def 4;
    then consider p being Real such that
A4: p > 0 and
A5: Ball(v,p) c= M by A1,Def5;
    v in N by A3,XBOOLE_0:def 4;
    then consider r being Real such that
A6: r > 0 and
A7: Ball(v,r) c= N by A2,Def5;
    take q = min(p,r);
    thus q > 0 by A4,A6,XXREAL_0:15;
    Ball(v,q) c= Ball(v,r) by Th33,XXREAL_0:17;
    then
A8: Ball(v,q) c= N by A7;
    Ball(v,q) c= Ball(v,p) by Th33,XXREAL_0:17;
    then Ball(v,q) c= M by A5;
    hence thesis by A8,XBOOLE_1:19;
  end;
  hence thesis by Def5;
end;
