
theorem
  for L be lower-bounded Semilattice for a,b,c be Element of L holds a
  <= b & a <= c & b"/\"c = Bottom L implies a = Bottom L
proof
  let L be lower-bounded Semilattice;
  let a,b,c be Element of L;
  assume that
A1: a <= b and
A2: a <= c and
A3: b"/\"c = Bottom L;
  a"/\"c <= b"/\"c by A1,Th6;
  then a <= b"/\"c by A2,Th10;
  hence thesis by A3,Th19;
end;
