
theorem
  for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
f be PartFunc of X,ExtREAL, c be Real, B be Element of S st f is_integrable_on
M holds f|B is_integrable_on M & Integral_on(M,B,c(#)f) =  c * Integral_on
  (M,B,f)
proof
  let X be non empty set, S be SigmaField of X, M be sigma_Measure of S, f be
  PartFunc of X,ExtREAL, c be Real, B be Element of S;
  assume f is_integrable_on M;
  then
A1: f|B is_integrable_on M by Th97;
A2: for x be object st x in dom((c(#)f)|B) holds (c(#)f)|B.x = (c(#)(f|B)).x
  proof
    let x be object;
    assume
A3: x in dom ((c(#)f)|B);
    then
A4: (c(#)f)|B.x= (c(#)f).x by FUNCT_1:47;
A5: x in dom (c(#)f) /\ B by A3,RELAT_1:61;
    then x in dom f /\ B by MESFUNC1:def 6;
    then
A6: x in dom (f|B) by RELAT_1:61;
    x in dom (c(#)f) by A5,XBOOLE_0:def 4;
    then (c(#)f)|B.x=  c * f.x by A4,MESFUNC1:def 6;
    then
A7: (c(#)f)|B.x=  c * f|B.x by A6,FUNCT_1:47;
    x in dom (c(#)(f|B)) by A6,MESFUNC1:def 6;
    hence thesis by A7,MESFUNC1:def 6;
  end;
  dom((c(#)f)|B) = dom(c(#)f) /\ B by RELAT_1:61;
  then dom((c(#)f)|B) = dom f /\ B by MESFUNC1:def 6;
  then dom((c(#)f)|B) = dom(f|B) by RELAT_1:61;
  then dom((c(#)f)|B) = dom(c(#)(f|B)) by MESFUNC1:def 6;
  then (c(#)f)|B = c(#)(f|B) by A2,FUNCT_1:2;
  hence thesis by A1,Th110;
end;
