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 Th49:
  for R being approximating auxiliary(i) auxiliary(iii) Relation of L holds
  [x,z] in R & x <> z implies ex y st x <= y & [y,z] in R & x <> y
proof
  let R be approximating auxiliary(i) auxiliary(iii) Relation of L;
  assume that
A1: [x,z] in R and
A2: x <> z;
  x <= z by A1,Def3;
  then x < z by A2,ORDERS_2:def 6;
  then not z < x by ORDERS_2:4;
  then not z <= x by A2,ORDERS_2:def 6;
  then consider u be Element of L such that
A3: [u,z] in R and
A4: not u <= x by Th48;
  take y = x "\/" u;
  thus x <= y by YELLOW_0:22;
  thus [y,z] in R by A1,A3,Def5;
  thus thesis by A4,YELLOW_0:24;
end;
