reserve a for set;

theorem Th12:
  for L being lower-bounded sup-Semilattice, x being Element of L
  for AR being auxiliary(i) Relation of L holds AR-below x c= downarrow x
proof
  let L be lower-bounded sup-Semilattice, x be Element of L,
  AR be auxiliary(i) Relation of L;
  let a be object;
  assume a in AR-below x;
  then consider y1 be Element of L such that
A1: y1 = a and
A2: [y1,x] in AR;
  y1 <= x by A2,Def3;
  hence thesis by A1,WAYBEL_0:17;
end;
