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
  A c= dom f & A c= dom g & f is_integrable_on A & f|A is bounded & g
  is_integrable_on A & g|A is bounded implies f(#)g is_integrable_on A
proof
  assume that
A1: A c= dom f & A c= dom g and
A2: f is_integrable_on A & f|A is bounded and
A3: g is_integrable_on A & g|A is bounded;
A4: f||A is integrable & f||A|A is bounded by A2;
A5: g||A is integrable & g||A|A is bounded by A3;
A6: (f||A)(#)(g||A) = (f(#)g)||A by INTEGRA5:4;
  f||A is Function of A,REAL & g||A is Function of A,REAL by A1,Lm1;
  then (f(#)g)||A is integrable by A4,A5,A6,INTEGRA4:29;
  hence thesis;
end;
