
theorem Th52:
  for W being with_non-empty_element set holds
  W-INF(SC)_category, W-SUP(SO)_category are_anti-isomorphic_under W LowerAdj
proof
  let W be with_non-empty_element set;
  set A1 = W-INF_category;
  set B1 = W-INF(SC)_category, B2 = W-SUP(SO)_category;
  set F = W LowerAdj;
A1: ex a being non empty set st a in W by SETFAM_1:def 10;
A2: for a being Object of A1 holds
  a is Object of B1 iff F.a is Object of B2 by Def10,Def11;
A3: now
    let a,b be Object of A1 such that
A4: <^a,b^> <> {};
    let a1,b1 be Object of B1 such that
A5: a1 = a and
A6: b1 = b;
    let a2,b2 be Object of B2 such that
A7: a2 = F.a and
A8: b2 = F.b;
    let f be Morphism of a,b;
A9: <^F.b,F.a^> <> {} by A4,FUNCTOR0:def 19;
A10: @f = f by A4,YELLOW21:def 7;
A11: @(F.f) = F.f by A9,YELLOW21:def 7;
A12: F.a = latt a by Def6;
A13: F.b = latt b by Def6;
A14: F.f = LowerAdj @f by A4,Def6;
    reconsider g = f as infs-preserving Function
    of latt a1, latt b1 by A1,A4,A5,A6,A10,Def4;
    UpperAdj LowerAdj g = g by Th10;
    then f in <^a1,b1^> iff UpperAdj LowerAdj g is directed-sups-preserving
    by A4,A5,A6,A10,Def10;
    hence f in <^a1,b1^> iff F.f in <^b2,a2^> by A5,A6,A7,A8,A9,A10,A11,A12,A13
,A14,Def11;
  end;
  thus thesis by A2,A3,YELLOW20:57;
end;
