reserve
  A,B,X for set,
  S for SigmaField of X;
reserve C for C_Measure of X;

theorem Th13:
  sigma_Meas(C) is Measure of sigma_Field(C)
proof
A1: now
    let A be Element of sigma_Field(C);
    reconsider A9 = A as Subset of X;
A2: C is nonnegative by Def1;
    (sigma_Meas(C)).A9 = C.A9 by Def3;
    hence 0.<= (sigma_Meas(C)).A by A2,MEASURE1:def 2;
  end;
  {} in sigma_Field(C) by PROB_1:4;
  then
A3: (sigma_Meas(C)).{} = C.{} by Def3;
A4: now
    let A,B be Element of sigma_Field(C);
    reconsider A9 = A,B9 = B as Subset of X;
A5: (sigma_Meas(C)).B9 = C.B9 by Def3;
    assume A misses B;
    then
A6: C.(A9 \/ B9) = C.A9 + C.B9 by Th6;
    (sigma_Meas(C)).A9 = C.A9 by Def3;
    hence (sigma_Meas(C)).(A \/ B) = (sigma_Meas(C)).A + (sigma_Meas(C)).B by
A5,A6,Def3;
  end;
  C is zeroed by Def1;
  then (sigma_Meas(C)).{} = 0. by A3,VALUED_0:def 19;
  hence thesis by A1,A4,MEASURE1:def 2,def 8,VALUED_0:def 19;
end;
