
theorem
for X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
 E be Element of S, r be Real holds
  Integral(M,r(#)chi(E,X)) = r * Integral(M,chi(E,X))
proof
   let X be non empty set, S be SigmaField of X, M be sigma_Measure of S,
   E be Element of S, r be Real;
A3:chi(E,X) is_simple_func_in S by Th12;
   Integral(M,r(#)chi(E,X)) = r * integral'(M,chi(E,X)) by Th12,MESFUN11:59;
   hence Integral(M,r(#)chi(E,X)) = r * Integral(M,chi(E,X)) by A3,MESFUNC5:89;
end;
