
theorem Th41:
  for L being lower-bounded LATTICE, R being auxiliary Relation of
L holds R is multiplicative iff for a,b,x,y being Element of L st [a,x] in R &
  [b,y] in R holds [a"/\"b,x"/\"y] in R
proof
  let L be lower-bounded LATTICE, R be auxiliary Relation of L;
  hereby
    assume
A1: R is multiplicative;
    let a,b,x,y be Element of L;
    assume that
A2: [a,x] in R and
A3: [b,y] in R;
    a"/\"b <= b & y <= y by YELLOW_0:23;
    then
A4: [a"/\"b,y] in R by A3,WAYBEL_4:def 4;
    a >= a"/\"b & x <= x by YELLOW_0:23;
    then [a"/\"b,x] in R by A2,WAYBEL_4:def 4;
    hence [a"/\"b,x"/\"y] in R by A1,A4;
  end;
  assume
A5: for a,b,x,y being Element of L st [a,x] in R & [b,y] in R holds [a
  "/\"b,x"/\"y] in R;
  let a be Element of L;
  a"/\"a = a by YELLOW_0:25;
  hence thesis by A5;
end;
