
theorem
  for L being lower-bounded LATTICE, R being auxiliary Relation of L
  holds R is multiplicative iff for x being Element of L holds R-above x is
  filtered
proof
  let L be lower-bounded LATTICE, R be auxiliary Relation of L;
  hereby
    assume
A1: R is multiplicative;
    let x be Element of L;
    thus R-above x is filtered
    proof
      let y,z be Element of L;
      assume y in R-above x & z in R-above x;
      then [x,y] in R & [x,z] in R by WAYBEL_4:14;
      then [x,y"/\"z] in R by A1;
      then
A2:   y"/\"z in R-above x by WAYBEL_4:14;
      y >= y"/\"z & z >= y"/\"z by YELLOW_0:23;
      hence thesis by A2;
    end;
  end;
  assume
A3: for x being Element of L holds R-above x is filtered;
  let a,x,y be Element of L;
  assume [a,x] in R & [a,y] in R;
  then
A4: x in R-above a & y in R-above a by WAYBEL_4:14;
  R-above a is filtered by A3;
  then x"/\"y in R-above a by A4,WAYBEL_0:41;
  hence [a,x"/\"y] in R by WAYBEL_4:14;
end;
