
theorem Th15:
  for L1, L2 being LATTICE st the RelStr of L1 = the RelStr of L2
  for x being set st x is prime Filter of L1 holds x is prime Filter of L2
proof
  let L1,L2 be LATTICE such that
A1: the RelStr of L1 = the RelStr of L2;
  let x be set;
  assume x is prime Filter of L1;
  then reconsider I = x as prime Filter of L1;
  reconsider I9 = I as Subset of L2 by A1;
  reconsider I9 as Filter of L2 by A1,WAYBEL_0:4,25;
  I9 is prime
  proof
    let x,y be Element of L2;
    reconsider a = x, b = y as Element of L1 by A1;
A2: x"\/"y = sup {x,y} by YELLOW_0:41;
    ex_sup_of {a,b}, L1 & a"\/"b = sup {a,b} by YELLOW_0:20,41;
    then a"\/"b = x"\/"y by A1,A2,YELLOW_0:26;
    hence thesis by Def2;
  end;
  hence thesis;
end;
