
theorem Th17:
  for W being with_non-empty_element set holds
  the carrier of W-INF_category = the carrier of W-SUP_category
proof
  let W be with_non-empty_element set;
A1: ex x being non empty set st x in W by SETFAM_1:def 10;
  thus the carrier of W-INF_category c= the carrier of W-SUP_category
  proof
    let x be object;
    assume
A2: x in the carrier of W-INF_category;
    then reconsider x as LATTICE by YELLOW21:def 4;
A3: x is strict complete by A1,A2,Def4;
    the carrier of x in W by A1,A2,Def4;
    then x is Object of W-SUP_category by A3,Def5;
    hence thesis;
  end;
  let x be object;
  assume
A4: x in the carrier of W-SUP_category;
  then reconsider x as LATTICE by YELLOW21:def 4;
A5: x is strict complete by A1,A4,Def5;
  the carrier of x in W by A1,A4,Def5;
  then x is Object of W-INF_category by A5,Def4;
  hence thesis;
end;
