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 Th9:
   A c= dom f & f is_integrable_on A & f|A is
  bounded implies
r(#)f is_integrable_on A & integral(r(#)f,A)=r*integral(f,A)
proof
  assume that
A1: A c= dom f and
A2: f is_integrable_on A & f|A is bounded;
A3: f||A is integrable & f||A|A is bounded by A2;
  rng f c= REAL;
  then f is Function of dom f,REAL by FUNCT_2:2;
  then
A4: f|A is Function of A,REAL by A1,FUNCT_2:32;
  r(#)(f||A) =(r(#)f)|A by RFUNCT_1:49;
  then (r(#)f)||A is integrable by A3,A4,INTEGRA2:31;
  hence r(#)f is_integrable_on A;
  integral((r(#)f),A) =integral(r(#)(f||A)) by RFUNCT_1:49;
  hence thesis by A3,A4,INTEGRA2:31;
end;
