theorem
  r <> 0 implies r ** (F/\G) = (r**F) /\ (r**G)
proof
  assume
A1: r <> 0;
A2: (r**F) /\ (r**G) c= r ** (F /\ G)
  proof
    let j;
    assume
A3: j in (r**F) /\ (r**G);
    then j in r**F by XBOOLE_0:def 4;
    then consider w such that
A4: j = r*w and
A5: w in F by Th188;
    j in r**G by A3,XBOOLE_0:def 4;
    then consider w1 such that
A6: j = r*w1 and
A7: w1 in G by Th188;
    w = w1 by A1,A4,A6,XXREAL_3:68;
    then w in F /\ G by A5,A7,XBOOLE_0:def 4;
    hence thesis by A4,Th186;
  end;
  r ** (F /\ G) c= (r**F) /\ (r**G) by Th83;
  hence thesis by A2;
end;
