
theorem
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
E be Element of S, f,g be PartFunc of X,ExtREAL st
 E c= dom f & E c= dom g & f is E-measurable & g
is E-measurable & f is nonpositive & (for x be Element of X st x in E holds
  g.x <= f.x ) holds Integral(M,g|E) <= Integral(M,f|E)
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    E be Element of S, f,g be PartFunc of X,ExtREAL;
    assume that
A1:  E c= dom f and
A2:  E c= dom g and
A3:  f is E-measurable & g is E-measurable and
A4:  f is nonpositive and
A5:  for x be Element of X st x in E holds g.x <= f.x;
    set f1 = -f, g1 = -g;
A6: E c= dom f1 & E c= dom g1 by A1,A2,MESFUNC1:def 7;
A7: f1 is E-measurable & g1 is E-measurable by A1,A2,A3,MEASUR11:63;
A11:for x be Element of X st x in E holds f1.x <= g1.x
    proof
     let x be Element of X;
     assume A9: x in E; then
     f1.x = -(f.x) & g1.x = -(g.x) by A6,MESFUNC1:def 7;
     hence f1.x <= g1.x by A5,A9,XXREAL_3:38;
    end;
A12:dom f /\ E = E & dom g /\ E = E by A1,A2,XBOOLE_1:28;
A14:E = dom(f|E) & E = dom(g|E) by A12,RELAT_1:61;
    f1|E = -(f|E) & g1|E = -(g|E) by Th3; then
    Integral(M,f1|E) = -Integral(M,f|E) & Integral(M,g1|E) = -Integral(M,g|E)
       by A3,A12,A14,Th52,MESFUNC5:42;
    hence Integral(M,g|E) <= Integral(M,f|E)
      by A4,A6,A7,A11,XXREAL_3:38,MESFUNC9:15;
end;
