
theorem Th20: :: PROPOSITION 4.7 b)
  for L be algebraic lower-bounded LATTICE holds L is arithmetic
  iff L-waybelow is multiplicative
proof
  let L be algebraic lower-bounded LATTICE;
  thus L is arithmetic implies L-waybelow is multiplicative
  proof
    assume
A1: L is arithmetic;
    now
      reconsider C = CompactSublatt L as meet-inheriting non empty full
      SubRelStr of L by A1;
      let a,x,y be Element of L;
      assume that
A2:   [a,x] in L-waybelow and
A3:   [a,y] in L-waybelow;
      a << x by A2,WAYBEL_4:def 1;
      then consider c be Element of L such that
A4:   c in the carrier of CompactSublatt L and
A5:   a <= c and
A6:   c <= x by Th7;
      a << y by A3,WAYBEL_4:def 1;
      then consider k be Element of L such that
A7:   k in the carrier of CompactSublatt L and
A8:   a <= k and
A9:   k <= y by Th7;
A10:  c"/\"k <= x"/\"y by A6,A9,YELLOW_3:2;
      reconsider c9=c, k9=k as Element of C by A4,A7;
      c9"/\"k9 in the carrier of CompactSublatt L;
      then c"/\"k in the carrier of CompactSublatt L by YELLOW_0:69;
      then c"/\"k is compact by Def1;
      then
A11:  c"/\"k << c"/\"k by WAYBEL_3:def 2;
      a"/\"a = a by YELLOW_5:2;
      then a <= c"/\"k by A5,A8,YELLOW_3:2;
      then a << x"/\"y by A10,A11,WAYBEL_3:2;
      hence [a,x"/\"y] in L-waybelow by WAYBEL_4:def 1;
    end;
    hence thesis by WAYBEL_7:def 7;
  end;
  assume
A12: L-waybelow is multiplicative;
  now
    let x,y be Element of L;
    assume that
A13: x in the carrier of CompactSublatt L and
A14: y in the carrier of CompactSublatt L and
    ex_inf_of {x,y},L;
    y is compact by A14,Def1;
    then y << y by WAYBEL_3:def 2;
    then
A15: [y,y] in L-waybelow by WAYBEL_4:def 1;
    x is compact by A13,Def1;
    then x << x by WAYBEL_3:def 2;
    then [x,x] in L-waybelow by WAYBEL_4:def 1;
    then [x "/\" y,x "/\" y] in L-waybelow by A12,A15,WAYBEL_7:41;
    then x "/\" y << x "/\" y by WAYBEL_4:def 1;
    then x "/\" y is compact by WAYBEL_3:def 2;
    then x "/\" y in the carrier of CompactSublatt L by Def1;
    hence inf {x,y} in the carrier of CompactSublatt L by YELLOW_0:40;
  end;
  then CompactSublatt L is meet-inheriting by YELLOW_0:def 16;
  hence thesis;
end;
