theorem Th137:
  r ++ (F /\ G) = (r++F) /\ (r++G)
proof
A1: (r++F) /\ (r++G) c= r ++ (F /\ G)
  proof
    let j;
    assume
A2: j in (r++F) /\ (r++G);
    then j in r++F by XBOOLE_0:def 4;
    then consider w such that
A3: j = r+w and
A4: w in F by Th134;
    j in r++G by A2,XBOOLE_0:def 4;
    then consider w1 such that
A5: j = r+w1 and
A6: w1 in G by Th134;
    w = w1 by A3,A5,XXREAL_3:11;
    then w in F /\ G by A4,A6,XBOOLE_0:def 4;
    hence thesis by A3,Th132;
  end;
  r ++ (F /\ G) c= (r++F) /\ (r++G) by Th42;
  hence thesis by A1;
end;
