
theorem
  for L1,L2 being RelStr st the RelStr of L1 = the RelStr of L2 & L1 is
  with_suprema holds L2 is with_suprema
proof
  let L1,L2 be RelStr such that
A1: the RelStr of L1 = the RelStr of L2 and
A2: for x,y being Element of L1 ex z being Element of L1 st z >= x & z
  >= y & for z9 being Element of L1 st z9 >= x & z9 >= y holds z9 >= z;
  let a,b be Element of L2;
  reconsider x = a, y = b as Element of L1 by A1;
  consider z being Element of L1 such that
A3: z >= x & z >= y and
A4: for z9 being Element of L1 st z9 >= x & z9 >= y holds z9 >= z by A2;
  reconsider c = z as Element of L2 by A1;
  take c;
  thus c >= a & c >= b by A1,A3;
  let d be Element of L2;
  reconsider z9 = d as Element of L1 by A1;
  assume d >= a & d >= b;
  then z9 >= x & z9 >= y by A1;
  then z9 >= z by A4;
  hence thesis by A1;
end;
