
theorem Th56:
  for H being complete non empty Poset holds H is Heyting iff H
  is meet-continuous distributive
proof
  let H be complete non empty Poset;
  hereby
    assume
A1: H is Heyting;
    then for x being Element of H holds x "/\" is lower_adjoint;
    then for X being Subset of H, x being Element of H holds x "/\" sup X =
    "\/"({x"/\"y where y is Element of H: y in X},H) by WAYBEL_1:64;
    then H is up-complete satisfying_MC by Th37;
    hence H is meet-continuous;
    thus H is distributive by A1;
  end;
  assume
A2: H is meet-continuous distributive;
  thus H is LATTICE;
  let a be Element of H;
  ( for X being finite Subset of H holds a "/\" preserves_sup_of X)& for X
being non empty directed Subset of H holds a "/\" preserves_sup_of X by A2,Th17
,WAYBEL_0:def 37;
  then a "/\" is sups-preserving by WAYBEL_0:74;
  hence thesis by WAYBEL_1:17;
end;
