reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;
reserve L for complete LATTICE;
reserve AR for Relation of L;
reserve x, y, z for Element of L;

theorem
  AR is satisfying_SI implies
  for x holds ( ex y st y is_maximal_wrt (AR-below x), AR )
  implies [x,x] in AR
proof
  assume
A1: AR is satisfying_SI;
  let x;
  given y such that
A2: y is_maximal_wrt (AR-below x), AR;
A3: y in AR-below x by A2;
  assume
A4: not [x,x] in AR;
A5: [y,x] in AR by A3,Th13;
  per cases;
  suppose x = y;
    hence contradiction by A3,A4,Th13;
  end;
  suppose x <> y;
    then consider z such that
A6: [y,z] in AR and
A7: [z,x] in AR and
A8: z <> y by A1,A5;
    z in AR-below x by A7;
    hence contradiction by A2,A6,A8;
  end;
end;
