theorem Th31:
  for V,W st V in Family_open_set(PM) & W in Family_open_set(PM)
  holds V /\ W in Family_open_set(PM)
proof
  let V,W;
  assume that
A1: V in Family_open_set(PM) and
A2: W in Family_open_set(PM);
  for z st z in V /\ W ex q st q>0 & Ball(z,q) c= V /\ W
  proof
    let z;
    assume
A3: z in V /\ W;
    then z in V by XBOOLE_0:def 4;
    then consider p such that
A4: p > 0 and
A5: Ball(z,p) c= V by A1,Def4;
    z in W by A3,XBOOLE_0:def 4;
    then consider r such that
A6: r > 0 and
A7: Ball(z,r) c= W by A2,Def4;
    take q = min(p,r);
    thus q > 0 by A4,A6,XXREAL_0:15;
    Ball(z,q) c= Ball(z,r) by Th1,XXREAL_0:17;
    then
A8: Ball(z,q) c= W by A7;
    Ball(z,q) c= Ball(z,p) by Th1,XXREAL_0:17;
    then Ball(z,q) c= V by A5;
    hence thesis by A8,XBOOLE_1:19;
  end;
  hence thesis by Def4;
end;
