reserve X for RealNormSpace;

theorem Th13:
  for r be Real
  for A be non empty closed_interval Subset of REAL
  for f be PartFunc of REAL,the carrier of X
    st A c= dom f & f is_integrable_on A holds
    r(#)f is_integrable_on A & integral((r(#)f),A) = r * integral(f,A)
proof
  let r be Real;
  let A be non empty closed_interval Subset of REAL;
  let f be PartFunc of REAL,the carrier of X;
  assume A1: A c= dom f & f is_integrable_on A;
A2: A c= dom(r(#)f) by A1,VFUNCT_1:def 4;
  consider g be Function of A,the carrier of X such that
A3: g = f|A & g is integrable by A1;
A4: (r(#)f)|A = r(#)(f|A) by VFUNCT_1:31
             .= r(#)g by A3,Th12;
  r(#)g is total by VFUNCT_1:34;then
  reconsider gg = r(#)g as Function of A,the carrier of X;
  gg is integrable & integral(gg) = r * integral(g) by A3,Th4;
  hence r(#)f is_integrable_on A by A4;
  thus integral((r(#)f),A) = integral(gg) by A4,A2,Def8
                          .= r * integral(g) by A3,Th4
                          .= r * integral(f,A) by A3,A1,Def8;
end;
