
theorem
for X be set, S be semialgebra_of_sets of X,
 P be pre-Measure of S, M be induced_Measure of S,P holds
  M = (C_Meas M)|(Field_generated_by S)
proof
   let X be set, S be semialgebra_of_sets of X,
   P be pre-Measure of S, M be induced_Measure of S,P;
   now let A be Element of Field_generated_by S;
    M is completely-additive by MEASURE9:60; then
    M.A = (C_Meas M).A by MEASURE8:18;
    hence M.A = ((C_Meas M)|(Field_generated_by S)).A by FUNCT_1:49;
   end;
   hence M = (C_Meas M)|(Field_generated_by S) by FUNCT_2:def 8;
end;
