
theorem
for X be non empty set, S be semialgebra_of_sets of X,
  P be pre-Measure of S, m be induced_Measure of S,P,
  M be induced_sigma_Measure of S,m holds M is_extension_of m
proof
   let X be non empty set, S be semialgebra_of_sets of X,
       P be pre-Measure of S, m be induced_Measure of S,P,
       M be induced_sigma_Measure of S,m;
   m is completely-additive by Th60; then
   consider N be sigma_Measure of sigma Field_generated_by S such that
A2: N is_extension_of m
  & N = (sigma_Meas(C_Meas m))|(sigma (Field_generated_by S)) by MEASURE8:33;
   thus M is_extension_of m by A2,Def10;
end;
