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 Th7:
  x in meet rng B iff for n being Nat holds x in B.n
proof
A1: dom B = NAT by FUNCT_2:def 1;
A2: now
    let x;
    assume
A3: for n being Nat holds x in B.n;
    now
      let Y;
      assume Y in rng B;
      then consider n be object such that
A4:   n in NAT and
A5:   Y = B.n by A1,FUNCT_1:def 3;
      thus x in Y by A3,A4,A5;
    end;
    hence x in meet rng B by SETFAM_1:def 1;
  end;
  now
    let x;
    assume
A6: x in meet rng B;
    now
      let k be Nat;
      k in NAT by ORDINAL1:def 12;
      then B.k in rng B by FUNCT_2:4;
      hence x in B.k by A6,SETFAM_1:def 1;
    end;
    hence for n being Nat holds x in B.n;
  end;
  hence thesis by A2;
end;
