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 Th52:
  for L being lower-bounded continuous LATTICE,
  x,y being Element of L st x << y
  ex x9 being Element of L st x << x9 & x9 << y
proof
  let L be lower-bounded continuous LATTICE;
  let x,y be Element of L;
  set R = L-waybelow;
  assume x << y;
  then [x,y] in R by Def1;
  then consider x9 be Element of L such that
A1: [x,x9] in R and
A2: [x9,y] in R by Def21;
A3: x << x9 by A1,Def1;
  x9 << y by A2,Def1;
  hence thesis by A3;
end;
