
theorem Th25:
  for L being complete transitive antisymmetric non empty RelStr
for A being Subset of L, B being non empty Subset of L holds A is_<=_than sup (
  A "\/" B)
proof
  let L be complete transitive antisymmetric non empty RelStr, A be Subset
  of L, B be non empty Subset of L;
  set b = the Element of B;
  let x be Element of L;
  assume x in A;
  then
A1: x "\/" b in A "\/" B;
  ex xx being Element of L st x <= xx & b <= xx & for c being Element of L
  st x <= c & b <= c holds xx <= c by LATTICE3:def 10;
  then
A2: x <= x "\/" b by LATTICE3:def 13;
  ex_sup_of A "\/" B,L by YELLOW_0:17;
  then A "\/" B is_<=_than sup (A "\/" B) by YELLOW_0:def 9;
  then x "\/" b <= sup (A "\/" B) by A1;
  hence thesis by A2,YELLOW_0:def 2;
end;
