
theorem Th59:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 f be PartFunc of X, ExtREAL, c be Real
 st f is_simple_func_in S & f is nonnegative
 holds Integral(M,c(#)f) = c * integral'(M,f)
proof
   let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
   f be PartFunc of X, ExtREAL, c be Real;
   assume that
A1: f is_simple_func_in S and
a2: f is nonnegative;
   per cases;
   suppose A2: c >= 0; then
A3: c(#)f is_simple_func_in S & c(#)f is nonnegative by A1,a2,MESFUNC5:20,39;
    integral'(M,c(#)f) = c * integral'(M,f) by A1,a2,A2,MESFUNC5:66;
    hence Integral(M,c(#)f) = c * integral'(M,f) by A3,MESFUNC5:89;
   end;
   suppose c < 0;
    hence Integral(M,c(#)f) = c * integral'(M,f) by A1,a2,Lm4;
   end;
end;
