theorem
  f is_integrable_on M implies r(#)f is_integrable_on M & Integral(M,r
  (#)f) = r * Integral(M,f)
proof
  assume f is_integrable_on M;
  then
A1: R_EAL f is_integrable_on M;
  then r(#)(R_EAL f) is_integrable_on M by MESFUNC5:110;
  then
A2: R_EAL(r(#)f) is_integrable_on M by Th20;
  Integral(M,r(#)(R_EAL f)) = r * Integral(M,R_EAL f)
      by A1,MESFUNC5:110;
  hence thesis by A2,Th20;
end;
