
theorem
  for L1,L2 being RelStr st the RelStr of L1 = the RelStr of L2 for X
  being set st ex_sup_of X,L1 holds "\/"(X,L1) = "\/"(X,L2)
proof
  let L1,L2 be RelStr such that
A1: the RelStr of L1 = the RelStr of L2;
  let X be set;
  reconsider c = "\/"(X,L1) as Element of L2 by A1;
  assume
A2: ex_sup_of X,L1;
  then X is_<=_than "\/"(X,L1) by Def9;
  then
A3: X is_<=_than c by A1,Th2;
A4: now
    let a be Element of L2;
    reconsider b = a as Element of L1 by A1;
    assume X is_<=_than a;
    then X is_<=_than b by A1,Th2;
    then b >= "\/"(X,L1) by A2,Def9;
    hence a >= c by A1;
  end;
  ex_sup_of X,L2 by A1,A2,Th14;
  hence thesis by A3,A4,Def9;
end;
