
theorem
  for L be lower-bounded antisymmetric transitive with_infima RelStr for
  a,b,c be Element of L holds a meets b & b <= c implies a meets c
proof
  let L be lower-bounded antisymmetric transitive with_infima RelStr;
  let a,b,c be Element of L;
  assume
A1: a meets b;
A2: Bottom L <= a"/\"b by YELLOW_0:44;
  assume b <= c;
  then
A3: a"/\"b <= a"/\"c by Th6;
  assume not a meets c;
  then not a"/\"c <> Bottom L;
  then a"/\"b = Bottom L by A3,A2,YELLOW_0:def 3;
  hence contradiction by A1;
end;
