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

theorem Th35: ::3.14 (1-3), p.71
  for L being lower-bounded continuous LATTICE holds L is
  distributive iff PRIME L is order-generating
proof
  let L be lower-bounded continuous LATTICE;
  thus L is distributive implies PRIME L is order-generating
  proof
    assume L is distributive;
    then
A1: PRIME L = IRR L by Th28;
    IRR L \ {Top L} c= IRR L by XBOOLE_1:36;
    hence thesis by A1,Th18,Th19;
  end;
  thus thesis by Th34;
end;
