
theorem Th24:
  for L be lower-bounded with_infima antisymmetric non empty
RelStr for x,y be Element of L st x is atom & y is atom & x <> y holds x "/\" y
  = Bottom L
proof
  let L be lower-bounded with_infima antisymmetric non empty RelStr;
  let x,y be Element of L;
  assume that
A1: x is atom and
A2: y is atom and
A3: x <> y and
A4: x "/\" y <> Bottom L;
  Bottom L <= x "/\" y by YELLOW_0:44;
  then
A5: Bottom L < x "/\" y by A4,ORDERS_2:def 6;
A6: x "/\" y <= y by YELLOW_0:23;
  x "/\" y <= x by YELLOW_0:23;
  then x = x "/\" y by A1,A5
    .= y by A2,A5,A6;
  hence contradiction by A3;
end;
