
theorem Th27:
  for X being non empty set,
      S being SigmaField of X, M being sigma_Measure of S,
      f being PartFunc of X,ExtREAL, r being Real
  st dom f in S & 0 <= r & (for x be object st x in dom f holds f.x = r)
    holds Integral(M,f) = r * M.(dom 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, r be Real;
   assume that
A1: dom f in S & 0 <= r and
A2: for x be object st x in dom f holds f.x = r;
   now let y be object;
    assume y in rng f; then
    consider x be object such that
A3:  x in dom f & y = f.x by FUNCT_1:def 3;
    y = r by A2,A3;
    hence y in REAL by XREAL_0:def 1;
   end; then
   rng f c= REAL; then
   reconsider g = f as PartFunc of X,REAL by RELSET_1:6;
A4:Integral(M,g) = r * M.(dom g) by A1,A2,MESFUNC6:97;
   f = R_EAL g by MESFUNC5:def 7;
   hence thesis by A4,MESFUNC6:def 3;
end;
