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 AR being auxiliary(ii) auxiliary(iii) Relation of L holds
  AR is satisfying_INT implies for x holds AR-below x is_directed_wrt AR
proof
  let AR be auxiliary(ii) auxiliary(iii) Relation of L;
  assume
A1: AR is satisfying_INT;
  let x,y,z;
  assume that
A2: y in AR-below x and
A3: z in AR-below x;
A4: [y,x] in AR by A2,Th13;
A5: [z,x] in AR by A3,Th13;
  consider y9 be Element of L such that
A6: [y,y9] in AR and
A7: [y9,x] in AR by A1,A4;
  consider z9 be Element of L such that
A8: [z,z9] in AR and
A9: [z9,x] in AR by A1,A5;
  take u = y9 "\/" z9;
  [u,x] in AR by A7,A9,Def5;
  hence u in AR-below x;
A10: y <= y;
  y9 <= u by YELLOW_0:22;
  hence [y,u] in AR by A6,A10,Def4;
A11: z <= z;
  z9 <= u by YELLOW_0:22;
  hence [z,u] in AR by A8,A11,Def4;
end;
