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

theorem Th7:
  A in sigma_Field(C) implies X \ A in sigma_Field(C)
proof
  assume
A1: A in sigma_Field(C);
  for W,Z being Subset of X holds W c= X \ A & Z c= X \ (X \ A) implies C.
  W + C.Z <= C.(W \/ Z)
  proof
    let W,Z be Subset of X;
    assume that
A2: W c= X \ A and
A3: Z c= X \ (X \ A);
    X \ (X \ A) = X /\ A by XBOOLE_1:48;
    then Z c= A by A1,A3,XBOOLE_1:28;
    hence thesis by A1,A2,Def2;
  end;
  hence thesis by Def2;
end;
