
theorem
  for L be lower-bounded antisymmetric transitive with_infima RelStr for
  a,b,c be Element of L holds a misses b or a misses c implies a misses b"/\"c
proof
  let L be lower-bounded antisymmetric transitive with_infima RelStr;
  let a,b,c be Element of L;
  assume
A1: a misses b or a misses c;
  per cases by A1;
  suppose
A2: a misses b;
    a"/\" (b"/\"c) = (a"/\"b)"/\"c by LATTICE3:16
      .= Bottom L "/\" c by A2
      .= Bottom L by Th25;
    hence thesis;
  end;
  suppose
A3: a misses c;
    a"/\" (b"/\"c) = (a"/\"c)"/\"b by LATTICE3:16
      .= Bottom L "/\" b by A3
      .= Bottom L by Th25;
    hence thesis;
  end;
end;
