
theorem Th66:
  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 st f is_simple_func_in S & f is
  nonnegative & 0 <= c holds integral'(M,c(#)f) = (c)*integral'(M,f)
proof
  let X be non empty set;
  let S be SigmaField of X;
  let M be sigma_Measure of S;
  let f be PartFunc of X,ExtREAL;
  let c be Real;
  assume that
A1: f is_simple_func_in S and
A2: f is nonnegative and
A3: 0 <= c;
  set g = c(#)f;
A5: dom g = dom f by MESFUNC1:def 6;
A6: for x be set st x in dom g holds g.x = ( c)*f.x by MESFUNC1:def 6;
  per cases;
  suppose
A7: dom g = {};
    then integral'(M,f) = 0 by A5,Def14;
    then c*integral'(M,f) = 0;
    hence thesis by A7,Def14;
  end;
  suppose
A8: dom g <> {};
    then
A9:  integral'(M,f) = integral(M,f) by A5,Def14;
    reconsider cc = c as R_eal by XXREAL_0:def 1;
    c in REAL by XREAL_0:def 1;
    then c < +infty by XXREAL_0:9;
    then integral(M,g) = cc*integral'(M,f)
     by A1,A3,A5,A2,A6,A8,MESFUNC4:6,A9;
   hence thesis by A8,Def14;
  end;
end;
