reserve Omega, F for non empty set,
  f for SetSequence of Omega,
  X,A,B for Subset of Omega,
  D for non empty Subset-Family of Omega,
  n,m for Element of NAT,
  h,x,y,z,u,v,Y,I for set;

theorem Th3:
  for Y being non empty set holds for x holds x c= meet Y iff for y
  being Element of Y holds x c= y
proof
  let Y be non empty set;
  let x;
  thus x c= meet Y implies for y be Element of Y holds x c= y
  by SETFAM_1:def 1;
  assume
A1: for y being Element of Y holds x c= y;
    let z be object;
    assume
A2: z in x;
    now
      let u;
      assume u in Y;
      then x c= u by A1;
      hence z in u by A2;
    end;
    hence thesis by SETFAM_1:def 1;
end;
