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 Th51:
  for L being lower-bounded continuous LATTICE
  holds L-waybelow is satisfying_SI
proof
  let L be lower-bounded continuous LATTICE;
  set R = L-waybelow;
  thus R is satisfying_SI
  proof
    let x,z be Element of L;
    assume that
A1: [x,z] in R and
A2: x <> z;
    x << z by A1,Def1;
    hence thesis by A2,Th50;
  end;
end;
