
theorem
  for X be set, S be set st S is SigmaRing of X & X in S holds
    S is SigmaField of X
proof
   let X be set, S be set;
   assume that
A1: S is SigmaRing of X and
A2: X in S;
   reconsider S1 = S as non empty Subset-Family of X by A1;
A3:S1 is diff-closed by A1;
P1:now let A be Subset of X;
    assume A in S1; then
    X \ A in S1 by A2,A3;
    hence A` in S1 by SUBSET_1:def 4;
   end;
   X c= X; then
   reconsider X1 = X as Subset of X;
   now let A be SetSequence of X;
    assume P2: rng A c= S1;
    now let a be object;
     assume a in rng Complement A; then
     consider n be Element of NAT such that
P3:   a = (Complement A).n by FUNCT_2:113;
     a = (A.n)` by P3,PROB_1:def 2; then
P4:  a = X \ A.n by SUBSET_1:def 4;
     A.n in rng A by FUNCT_2:4;
     hence a in S1 by P4,P2,A1,A2,FINSUB_1:def 3;
    end; then
    rng Complement A c= S1; then
    Union Complement A in S1 by A1,DefSigmaRing; then
    X1 \ (Union Complement A) in S1 by A1,A2,FINSUB_1:def 3; then
    (Union Complement A)` in S1 by SUBSET_1:def 4;
    hence Intersection A in S1 by PROB_1:def 3;
   end;
   hence S is SigmaField of X by P1,PROB_1:def 1,def 6;
end;
