
theorem Th27:
  for L being with_suprema transitive antisymmetric RelStr for a,
b being Element of L, A, B being Subset of L st a is_>=_than A & b is_>=_than B
  holds a "\/" b is_>=_than A "\/" B
proof
  let L be with_suprema transitive antisymmetric RelStr, a, b be Element of L,
  A, B be Subset of L such that
A1: a is_>=_than A & b is_>=_than B;
  let q be Element of L;
  assume q in A "\/" B;
  then consider x, y being Element of L such that
A2: q = x "\/" y and
A3: x in A & y in B;
  a >= x & b >= y by A1,A3;
  hence thesis by A2,YELLOW_3:3;
end;
