
theorem Th49:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 E be Element of S, er be ExtReal holds
  Integral(M,chi(er,E,X)) = er * M.E
proof
   let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
   E be Element of S, er be ExtReal;
   reconsider XX = X as Element of S by MEASURE1:7;
   per cases;
   suppose a1: er = +infty; then
a2: chi(er,E,X) = Xchi(E,X) by Th2;
    per cases;
    suppose a3: M.E <> 0; then
a4:  M.E > 0 by MEASURE1:def 2;
     thus Integral(M,chi(er,E,X)) = +infty by a2,a3,MEASUR10:33
      .= er * M.E by a1,a4,XXREAL_3:def 5;
    end;
    suppose a5: M.E = 0; then
     Integral(M,chi(er,E,X)) = 0 by a2,MEASUR10:33;
     hence Integral(M,chi(er,E,X)) = er * M.E by a5;
    end;
   end;
   suppose a6: er = -infty; then
a7: chi(er,E,X) = -Xchi(E,X) by Th2;
a10:dom(Xchi(E,X)) = XX by FUNCT_2:def 1;
W:  Xchi(E,X) is XX-measurable by MEASUR10:32;
    per cases;
    suppose a8: M.E <> 0; then
a9:  M.E > 0 by MEASURE1:def 2;
     thus Integral(M,chi(er,E,X))
      = -Integral(M,Xchi(E,X)) by a10,a7,MESFUN11:52,W
     .= -(+infty) by a8,MEASUR10:33
     .= er * M.E by a6,a9,XXREAL_3:def 5,6;
    end;
    suppose a12: M.E = 0;
     thus Integral(M,chi(er,E,X))
      = - Integral(M,Xchi(E,X)) by a10,a7,MESFUN11:52,W
     .= - 0 by a12,MEASUR10:33
     .= er * M.E by a12;
    end;
   end;
   suppose er <> +infty & er <> -infty; then
    er in REAL by XXREAL_0:14; then
    reconsider r = er as Real;
a14:chi(E,X) is_simple_func_in S by Th12;
    chi(er,E,X) = r(#)chi(E,X) by Th1;
    hence Integral(M,chi(er,E,X))
     = r * integral'(M,chi(E,X)) by Th12,MESFUN11:59
    .= r * Integral(M,chi(E,X)) by a14,MESFUNC5:89
    .= er * M.E by MESFUNC9:14;
   end;
end;
