
theorem
  for L1, L2 being LATTICE st the RelStr of L1 = the RelStr of L2 for x
  being set st x is prime Ideal of L1 holds x is prime Ideal 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 Ideal of L1;
  then reconsider I = x as prime Ideal of L1;
  reconsider I9 = I as Subset of L2 by A1;
  reconsider I9 as Ideal of L2 by A1,WAYBEL_0:3,25;
  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 = inf {x,y} by YELLOW_0:40;
    ex_inf_of {a,b}, L1 & a"/\"b = inf {a,b} by YELLOW_0:21,40;
    then a"/\"b = x"/\"y by A1,A2,YELLOW_0:27;
    hence thesis by Def1;
  end;
  hence thesis;
end;
