
theorem
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 nonpositive
 holds Integral(M,c(#)f) = (-c) * integral'(M,-f)
     & 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 nonpositive;
   set d = -c;
A3:d(#)(-f) = d(#)((-1)(#)f) by MESFUNC2:9 .= (d*(-1))(#)f by Th35;
   hence Integral(M,c(#)f) = (-c) * integral'(M,-f) by A2,A1,Th30,Th59;
   Integral(M,c(#)f) = d * integral'(M,-f) by A2,A1,A3,Th30,Th59;
   hence Integral(M,c(#)f) = -(c * integral'(M,-f)) by XXREAL_3:92;
end;
