reserve a for set;
reserve L for lower-bounded sup-Semilattice;
reserve x for Element of L;

theorem Th29:
  for R1, R2 being Relation of L st R1 c= R2 holds R1-below x c= R2-below x
proof
  let R1, R2 be Relation of L;
  assume
A1: R1 c= R2;
  let a be object;
  assume a in R1-below x;
  then ex b be Element of L st ( b = a)&( [b,x] in R1);
  hence thesis by A1;
end;
