reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,COMPLEX,
  r for Real,
  c for Complex,
  E,A,B for Element of S;

theorem
  f is_integrable_on M implies Integral_on(M,B,c(#)f) = c * Integral_on( M,B,f)
proof
  assume f is_integrable_on M;
  then
A1: f|B is_integrable_on M by Th23;
A2: dom((c(#)f)|B) = dom(c(#)f) /\ B by RELAT_1:61;
  then dom((c(#)f)|B) = dom f /\ B by VALUED_1:def 5;
  then
A3: dom((c(#)f)|B) = dom(f|B) by RELAT_1:61;
A4: now
    let x be object;
    assume
A5: x in dom((c(#)f)|B);
    then
A6: (c(#)f)|B.x= (c(#)f).x by FUNCT_1:47;
    x in dom (c(#)f) by A2,A5,XBOOLE_0:def 4;
    then (c(#)f)|B.x= c * f.x by A6,VALUED_1:def 5;
    then
A7: (c(#)f)|B.x= c * f|B.x by A3,A5,FUNCT_1:47;
    x in dom (c(#)(f|B)) by A3,A5,VALUED_1:def 5;
    hence (c(#)f)|B.x= (c(#)(f|B)).x by A7,VALUED_1:def 5;
  end;
  dom((c(#)f)|B) = dom(c(#)(f|B)) by A3,VALUED_1:def 5;
  then (c(#)f)|B = c(#)(f|B) by A4,FUNCT_1:2;
  hence thesis by A1,Th40;
end;
