
theorem Th16:
  for L being LATTICE, x being set holds x is prime Ideal of L iff
  x is prime Filter of L opp
proof
  let L be LATTICE, x be set;
  hereby
    assume x is prime Ideal of L;
    then reconsider I = x as prime Ideal of L;
    reconsider F = I as Filter of L opp by YELLOW_7:26,28;
    F is prime
    proof
      let x,y be Element of L opp;
A1:   x"\/"y = (~x)"/\"(~y) by YELLOW_7:22;
      ~x = x & ~y = y by LATTICE3:def 7;
      hence thesis by A1,Def1;
    end;
    hence x is prime Filter of L opp;
  end;
  assume x is prime Filter of L opp;
  then reconsider I = x as prime Filter of L opp;
  reconsider F = I as Ideal of L by YELLOW_7:26,28;
  F is prime
  proof
    let x,y be Element of L;
A2: x"/\"y = (x~)"\/"(y~) by YELLOW_7:21;
    x~ = x & y~ = y by LATTICE3:def 6;
    hence thesis by A2,Def2;
  end;
  hence thesis;
end;
