
theorem Th2:
  for L being complete LATTICE, X,Y being Subset of L st X is_finer_than Y
  holds "\/"(X,L) <= "\/"(Y,L)
proof
  let L be complete LATTICE, X,Y be Subset of L such that
A1: X is_finer_than Y;
  "\/"(Y,L) is_>=_than X
  proof
    let x be Element of L;
    assume x in X;
    then consider y being Element of L such that
A2: y in Y and
A3: x <= y by A1;
    "\/"(Y,L) >= y by A2,YELLOW_2:22;
    hence thesis by A3,YELLOW_0:def 2;
  end;
  hence thesis by YELLOW_0:32;
end;
