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 Th48:
  for AR being approximating Relation of L st
  not x <= y holds ex u being Element of L st [u,x] in AR & not u <= y
proof
  let AR be approximating Relation of L;
  assume
A1: not x <= y;
A2: x = sup (AR-below x) by Def17;
  ex_sup_of (AR-below x),L by YELLOW_0:17;
  then y is_>=_than (AR-below x) implies y >= x by A2,YELLOW_0:def 9;
  then consider u being Element of L such that
A3: u in AR-below x and
A4: not u <= y by A1;
  take u;
  thus thesis by A3,A4,Th13;
end;
