reserve X for non empty set,
  S for SigmaField of X,
  M for sigma_Measure of S,
  E for Element of S,
  F,G for Functional_Sequence of X,ExtREAL,
  I for ExtREAL_sequence,
  f,g for PartFunc of X,ExtREAL,
  seq, seq1, seq2 for ExtREAL_sequence,
  p for ExtReal,
  n,m for Nat,
  x for Element of X,
  z,D for set;

theorem Th15:
  E c= dom f & E c= dom g & f is E-measurable & g
is E-measurable & f is nonnegative & (for x be Element of X st x in E holds
  f.x <= g.x) implies Integral(M,f|E) <= Integral(M,g|E)
proof
  assume that
A1: E c= dom f and
A2: E c= dom g and
A3: f is E-measurable and
A4: g is E-measurable and
A5: f is nonnegative and
A6: for x be Element of X st x in E holds f.x <= g.x;
  set F2 = g|E;
A7: E = dom(f|E) by A1,RELAT_1:62;
  set F1 = f|E;
A8: F1 is nonnegative by A5,MESFUNC5:15;
A9: E = dom(g|E) by A2,RELAT_1:62;
A10: for x be Element of X st x in dom F1 holds F1.x <= F2.x
  proof
    let x be Element of X;
    assume
A11: x in dom F1;
    then
A12: F1.x = f.x by FUNCT_1:47;
    F2.x = g.x by A7,A9,A11,FUNCT_1:47;
    hence thesis by A6,A7,A11,A12;
  end;
  for x be object st x in dom F2 holds 0 <= F2.x
  proof
    let x be object;
    assume
A13: x in dom F2;
    0 <= F1.x by A8,SUPINF_2:51;
    hence thesis by A7,A9,A10,A13;
  end;
  then
A14: F2 is nonnegative by SUPINF_2:52;
A15: dom g /\ E = E by A2,XBOOLE_1:28;
  then
A16: F2 is E-measurable by A4,MESFUNC5:42;
A17: dom f /\ E = E by A1,XBOOLE_1:28;
  then F1 is E-measurable by A3,MESFUNC5:42;
  then integral+(M,F1) <= integral+(M,F2) by A8,A7,A9,A10,A14,A16,MESFUNC5:85;
  then Integral(M,F1) <= integral+(M,F2) by A3,A8,A7,A17,MESFUNC5:42,88;
  hence thesis by A4,A9,A14,A15,MESFUNC5:42,88;
end;
