reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  F for sequence of S,
  f,g for PartFunc of X,REAL,
  A,B for Element of S,
  r,s for Real,
  a for Real,
  n for Nat;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S,
  r for Real,
  p for Rational;
reserve X for non empty set,
  f,g for PartFunc of X,REAL,
  r for Real ;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL,
  A for Element of S;
reserve X for non empty set,
  Y for set,
  S for SigmaField of X,
  f,g,h for PartFunc of X,REAL,
  A for Element of S,
  r for Real;
reserve X for non empty set,
  S for SigmaField of X,
  f,g for PartFunc of X,REAL ,
  A,B for Element of S,
  r,s for Real;
reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  f,g for PartFunc of X,REAL,
  r for Real,
  E,A,B for Element of S;

theorem
  dom f in S & 0 <= r & (for x be object st x in dom f holds f.x = r)
  implies Integral(M,f) = r * M.(dom f)
proof
  assume that
A1: dom f in S and
A2: 0 <= r and
A3: for x be object st x in dom f holds f.x = r;
A4: for x be object st x in dom R_EAL f holds 0. <= (R_EAL f).x by A2,A3;
  reconsider A = dom R_EAL f as Element of S by A1;
A5: R_EAL f is A-measurable by A3,Th2,MESFUNC2:34;
  r * M.(dom R_EAL f) = integral'(M,R_EAL f) & R_EAL f
  is_simple_func_in S by A1,A2,A3,Th2,MESFUNC5:104;
  then integral+(M,R_EAL f) = r * M.(dom R_EAL f) by A4,MESFUNC5:77
,SUPINF_2:52;
  hence thesis by A4,A5,MESFUNC5:88,SUPINF_2:52;
end;
