reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;

theorem Th40:
  for L being complete LATTICE, x being Element of L holds
  (L-waybelow)-below x = waybelow x
proof
  let L be complete LATTICE, x be Element of L;
  set AR = L-waybelow;
  set A = {y where y is Element of L: [y,x] in AR};
  set B = {y where y is Element of L: y << x};
A1: A c= B
  proof
    let a be object;
    assume a in A;
    then consider v being Element of L such that
A2: a = v and
A3: [v,x] in AR;
    v << x by A3,Def1;
    hence thesis by A2;
  end;
  B c= A
  proof
    let a be object;
    assume a in B;
    then consider v being Element of L such that
A4: a = v and
A5: v << x;
    [v,x] in AR by A5,Def1;
    hence thesis by A4;
  end;
  then A = B by A1;
  hence thesis by WAYBEL_3:def 3;
end;
