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
  ( for x holds AR-below x is_directed_wrt AR) implies AR is satisfying_INT
proof
  assume
A1: for x holds AR-below x is_directed_wrt AR;
  let X,Z be Element of L;
  assume [X,Z] in AR;
  then
A2: X in AR-below Z;
  AR-below Z is_directed_wrt AR by A1;
  then consider u be Element of L such that
A3: u in AR-below Z and
A4: [X,u] in AR and [X,u] in AR by A2;
  take u;
  thus [X,u] in AR by A4;
  thus [u,Z] in AR by A3,Th13;
end;
