
theorem
  for L being with_suprema transitive antisymmetric RelStr for a, b, c,
  d being Element of L st a <= c & b <= d holds a "\/" b <= c "\/" d
proof
  let L be with_suprema transitive antisymmetric RelStr, a, b, c, d be Element
  of L such that
A1: a <= c and
A2: b <= d;
A3: ex_sup_of {c,d},L by YELLOW_0:20;
  then d <= c "\/" d by YELLOW_0:18;
  then
A4: b <= c "\/" d by A2,ORDERS_2:3;
  c <= c "\/" d by A3,YELLOW_0:18;
  then
  ( ex x being Element of L st a <= x & b <= x & for z being Element of L
  st a <= z & b <= z holds x <= z)& a <= c "\/" d by A1,LATTICE3:def 10
,ORDERS_2:3;
  hence thesis by A4,LATTICE3:def 13;
end;
