
theorem
  for L being lower-bounded LATTICE, R being auxiliary Relation of L
  holds R is multiplicative iff R-below is meet-preserving
proof
  let L be lower-bounded LATTICE, R be auxiliary Relation of L;
  hereby
    assume
A1: R is multiplicative;
    thus R-below is meet-preserving
    proof
      let x,y be Element of L;
A2:   (R-below).y = R-below y by WAYBEL_4:def 12;
A3:   R-below (x"/\"y) = (R-below x) /\ (R-below y)
      proof
        hereby
          let a be object;
          assume a in R-below (x"/\"y);
          then a in {z where z is Element of L: [z,x"/\"y] in R} by
WAYBEL_4:def 10;
          then consider z being Element of L such that
A4:       a = z and
A5:       [z,x"/\"y] in R;
A6:       z <= z;
          x"/\"y <= y by YELLOW_0:23;
          then [z,y] in R by A5,A6,WAYBEL_4:def 4;
          then
A7:       z in R-below y by WAYBEL_4:13;
          x"/\"y <= x by YELLOW_0:23;
          then [z,x] in R by A5,A6,WAYBEL_4:def 4;
          then z in R-below x by WAYBEL_4:13;
          hence a in (R-below x) /\ (R-below y) by A4,A7,XBOOLE_0:def 4;
        end;
        let a be object;
        assume
A8:     a in (R-below x) /\ (R-below y);
        then reconsider z = a as Element of L;
        a in R-below y by A8,XBOOLE_0:def 4;
        then
A9:     [z,y] in R by WAYBEL_4:13;
        a in R-below x by A8,XBOOLE_0:def 4;
        then [z,x] in R by WAYBEL_4:13;
        then [z,x"/\"y] in R by A1,A9;
        hence thesis by WAYBEL_4:13;
      end;
      (R-below).(x"/\"y) = R-below (x"/\"y) & (R-below).x = R-below x by
WAYBEL_4:def 12;
      then (R-below).(x"/\"y) = (R-below).x"/\"(R-below).y by A2,A3,YELLOW_2:43
;
      hence thesis by YELLOW_3:8;
    end;
  end;
  assume
A10: for x,y being Element of L holds R-below preserves_inf_of {x,y};
  let a,x,y be Element of L;
  R-below preserves_inf_of {x,y} by A10;
  then
A11: (R-below).(x"/\"y) = (R-below).x"/\"(R-below).y by YELLOW_3:8
    .= (R-below).x /\ (R-below).y by YELLOW_2:43;
A12: (R-below).y = R-below y by WAYBEL_4:def 12;
  assume [a,x] in R & [a,y] in R;
  then
A13: a in R-below x & a in R-below y by WAYBEL_4:13;
  (R-below).(x"/\"y) = R-below (x"/\"y) & (R-below).x = R-below x by
WAYBEL_4:def 12;
  then a in R-below (x"/\"y) by A11,A12,A13,XBOOLE_0:def 4;
  hence thesis by WAYBEL_4:13;
end;
