
theorem
  for L being lower-bounded LATTICE, R being auxiliary Relation of L
  holds R is multiplicative iff for S being full SubRelStr of [:L,L:] st the
  carrier of S = R holds S is meet-inheriting
proof
  let L be lower-bounded LATTICE, R be auxiliary Relation of L;
  reconsider X = R as Subset of [:L,L:] by YELLOW_3:def 2;
A1: X = the carrier of subrelstr X by YELLOW_0:def 15;
  hereby
    assume
A2: R is multiplicative;
    let S be full SubRelStr of [:L,L:] such that
A3: the carrier of S = R;
    thus S is meet-inheriting
    proof
      let x,y be Element of [:L,L:];
      assume
A4:   x in the carrier of S & y in the carrier of S;
      the carrier of [:L,L:] = [:the carrier of L, the carrier of L:] by
YELLOW_3:def 2;
      then
A5:   x = [x`1,x`2] & y = [y`1,y`2] by MCART_1:21;
      ex_inf_of {x,y}, [:L,L:] by YELLOW_0:21;
      then inf {x,y} = [inf proj1 {x,y}, inf proj2 {x,y}] by YELLOW_3:47
        .= [inf {x`1,y`1}, inf proj2 {x,y}] by A5,FUNCT_5:13
        .= [inf {x`1,y`1}, inf {x`2,y`2}] by A5,FUNCT_5:13
        .= [x`1"/\"y`1, inf {x`2,y`2}] by YELLOW_0:40
        .= [x`1"/\"y`1, x`2"/\"y`2] by YELLOW_0:40;
      hence thesis by A2,A3,A4,A5,Th41;
    end;
  end;
  assume for S being full SubRelStr of [:L,L:] st the carrier of S = R holds
  S is meet-inheriting;
  then
A6: subrelstr X is meet-inheriting by A1;
  let a,x,y be Element of L;
A7: ex_inf_of {[a,x],[a,y]}, [:L,L:] by YELLOW_0:21;
  assume [a,x] in R & [a,y] in R;
  then inf {[a,x],[a,y]} in R by A1,A6,A7;
  then
A8: [a,x]"/\"[a,y] in R by YELLOW_0:40;
  set ax = [a,x], ay = [a,y];
  [a,x]"/\"[a,y] = inf {ax,ay} by YELLOW_0:40
    .= [inf proj1 {ax,ay}, inf proj2 {ax,ay}] by A7,YELLOW_3:47
    .= [inf {a,a}, inf proj2 {ax,ay}] by FUNCT_5:13
    .= [inf {a,a}, inf {x,y}] by FUNCT_5:13
    .= [a"/\"a, inf {x,y}] by YELLOW_0:40
    .= [a"/\"a, x"/\"y] by YELLOW_0:40;
  hence [a,x"/\"y] in R by A8,YELLOW_0:25;
end;
