reserve x,y,Y,Z for set,
  L for LATTICE,
  l for Element of L;

theorem Th28:
  for L being distributive LATTICE holds PRIME L = IRR L
proof
  let L be distributive LATTICE;
  now
    let l1 be object;
    assume
A1: l1 in PRIME L;
    then reconsider l = l1 as Element of PRIME L;
    reconsider l as Element of L by A1;
    l is prime by A1,Def7;
    then l is irreducible by Th27;
    hence l1 in IRR L by Def4;
  end;
  hence PRIME L c= IRR L;
  now
    let l1 be object;
    assume
A2: l1 in IRR L;
    then reconsider l = l1 as Element of IRR L;
    reconsider l as Element of L by A2;
    l is irreducible by A2,Def4;
    then l is prime by Th27;
    hence l1 in PRIME L by Def7;
  end;
  hence thesis;
end;
