reserve n,m,k,k1,k2,i,j for Nat;
reserve x,y,z for object,X,Y,Z for set;
reserve A for Subset of X;
reserve B,A1,A2,A3 for SetSequence of X;
reserve Si for SigmaField of X;
reserve S,S1,S2,S3 for SetSequence of Si;

theorem Th8:
  Intersection B = meet rng B
proof
  now
    let x be object;
    assume x in meet rng B;
    then for n being Nat holds x in B.n by Th7;
    hence x in Intersection B by PROB_1:13;
  end;
  then
A1: meet rng B c= Intersection B;
  Intersection B c= meet rng B
  proof
    let x be object;
    assume x in Intersection B;
    then for n being Nat holds x in B.n by PROB_1:13;
    hence thesis by Th7;
  end;
  hence thesis by A1,XBOOLE_0:def 10;
end;
