
theorem Th32: :: THEOREM 4.23 (i)
  for L be lower-bounded algebraic LATTICE holds Irr L is
order-generating & for X be Subset of L st X is order-generating holds Irr L c=
  X
proof
  let L be lower-bounded algebraic LATTICE;
  now
    let x,y be Element of L;
    assume not y <= x;
    then consider p be Element of L such that
A1: p is completely-irreducible and
A2: x <= p and
A3: not y <= p by Th31;
    take p;
    thus p in Irr L by A1,Def4;
    thus x <= p by A2;
    thus not y <= p by A3;
  end;
  hence Irr L is order-generating by WAYBEL_6:17;
  let X be Subset of L;
  assume X is order-generating;
  hence thesis by Th25;
end;
