reserve a for set;

theorem
  for L being lower-bounded sup-Semilattice,
  AR being auxiliary(i) Relation of L for x being Element of L holds
  AR = the InternalRel of L implies AR-below x = downarrow x
proof
  let L be lower-bounded sup-Semilattice, AR be auxiliary(i) Relation of L;
  let x be Element of L;
  assume
A1: AR = the InternalRel of L;
  thus AR-below x c= downarrow x by Th12;
  thus downarrow x c= AR-below x
  proof
    let a be object;
    assume a in downarrow x;
    then [a,x] in AR by A1,Lm3;
    hence thesis by Th13;
  end;
end;
