theorem
  for X,S,f,g,A,r st f is A-measurable & g is A-measurable & A c= dom g
  holds A /\ less_dom(f,r) /\ great_dom(g,r) in S
proof
  let X,S,f,g,A,r;
  assume f is A-measurable & g is A-measurable & A c= dom g;
  then A1: A
 /\ less_dom(f,r) in S & A /\ great_dom(g,r) in S by Th29;
 (A /\ less_dom(f,r)) /\ (A /\ great_dom(g,r))
  =((A /\ less_dom(f,r)) /\ A) /\ great_dom(g,r) by XBOOLE_1:16
    .=((A /\ A) /\ less_dom(f,r)) /\ great_dom(g,r) by XBOOLE_1:16
    .=A /\ less_dom(f,r) /\ great_dom(g,r);
  hence thesis by A1,FINSUB_1:def 2;
end;
