
theorem
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
E be Element of S, f be PartFunc of X,ExtREAL st
 E = dom f & f is E-measurable & f is nonpositive & M.(E /\
 eq_dom(f,-infty)) <> 0 holds Integral(M,f) = -infty
proof
    let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
    E be Element of S, f be PartFunc of X,ExtREAL;
    assume that
A1:  E = dom f and
A2:  f is E-measurable and
A3:  f is nonpositive and
A4:  M.(E /\ eq_dom(f,-infty)) <> 0;
    set g = -f;
A5: E = dom g by A1,MESFUNC1:def 7;
    g = (-1)(#)f by MESFUNC2:9; then
    eq_dom(f,-infty) = eq_dom(g,-infty*(-1)) by Th9; then
    eq_dom(f,-infty) = eq_dom(g,+infty) by XXREAL_3:def 5; then
    Integral(M,g) = +infty by A1,A2,A3,A4,A5,MESFUNC9:13,MEASUR11:63; then
    -Integral(M,f) = +infty by A1,A2,Th52;
    hence Integral(M,f) = -infty by XXREAL_3:6;
end;
