
theorem Th3:
  for X being set, F being SetSequence of X, x being object
   holds x in meet F iff for z being Nat holds x in F.z
proof
  let X be set, F be SetSequence of X, x be object;
  hereby
    assume
A1: x in meet F;
    let z be Nat;
    z in NAT by ORDINAL1:def 12;
    then z in dom F by FUNCT_2:def 1;
    hence x in F.z by A1,FUNCT_6:25;
  end;
  assume for z being Nat holds x in F.z;
  then for y being object st y in dom F holds x in F.y;
  hence thesis by FUNCT_6:25;
end;
