theorem
  f is_integrable_on M implies f|B is_integrable_on M & Integral_on(M,B,
  r(#)f) = r * Integral_on(M,B,f)
proof
  assume f is_integrable_on M;
  then
A1: R_EAL f is_integrable_on M;
  then Integral_on(M,B,r(#)(R_EAL f)) = r * Integral_on(M,B,R_EAL f) by
MESFUNC5:112;
  then
A2: Integral_on(M,B,R_EAL(r(#)f)) = r * Integral_on(M,B,R_EAL f) by Th20;
  R_EAL(f|B) is_integrable_on M by A1,MESFUNC5:112;
  hence thesis by A2;
end;
