
theorem Th98:
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 A be Element of S, r be Real st r >= 0 holds
  Integral(M,r(#)chi(A,X)) = r * M.A
proof
   let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
   A be Element of S, r be Real;
   assume A1: r >= 0;
   reconsider XX = X as Element of S by MEASURE1:7;
A2:dom(chi(A,X)) = XX & chi(A,X) is XX-measurable
     by FUNCT_3:def 3,MESFUNC2:29; then
A3:dom(r(#)chi(A,X)) = XX & r(#)chi(A,X) is XX-measurable
     by MESFUNC1:def 6,37;
   Integral(M,chi(A,X)) = M.A by MESFUNC9:14; then
   integral+(M,chi(A,X)) = M.A by A2,MESFUNC5:88; then
   integral+(M,r(#)chi(A,X)) = r * M.A by A1,A2,MESFUNC5:86;
   hence Integral(M,r(#)chi(A,X)) = r * M.A by A1,A3,MESFUNC5:20,88;
end;
