reserve a for set;

theorem Th17:
  for L being lower-bounded sup-Semilattice,
  AR being auxiliary(ii) Relation of L for x, y being Element of L holds
  x <= y implies AR-below x c= AR-below y
proof
  let L be lower-bounded sup-Semilattice, AR be auxiliary(ii) Relation of L;
  let x, y be Element of L;
  assume
A1: x <= y;
  let a be object;
  assume a in AR-below x;
  then consider y2 be Element of L such that
A2: y2 = a and
A3: [y2,x] in AR;
  y2 <= y2;
  then [y2,y] in AR by A1,A3,Def4;
  hence thesis by A2;
end;
