reserve a,b,c,d,e,x,r for Real,
  A for non empty closed_interval Subset of REAL,
  f,g for PartFunc of REAL,REAL;

theorem
  f|A is bounded & g|A is bounded implies (f(#)g)|A is bounded
proof
  assume f|A is bounded & g|A is bounded;
  then (f(#)g)|(A /\ A) is bounded by RFUNCT_1:84;
  hence thesis;
end;
