
theorem Th59:
  for X being non empty finite set, F being Dependency-set of X
  holds charact_set F = charact_set (Dependency-closure F)
proof
  let X be non empty finite set, F be Dependency-set of X;
  set dcF = Dependency-closure F;
  now
    set B = {c where c is Subset of X : for x, y being Subset of X st [x, y]
    in F & x c= c holds y c= c};
    let A be object;
    reconsider AA=A as set by TARSKI:1;
    hereby
A1:   F c= dcF by Def24;
      assume
A2:   A in charact_set F;
      then
A3:   A is Subset of X by Th55;
      ex x, y being Subset of X st [x, y] in F & x c= AA & not y c= AA
      by A2,Th55;
      hence A in charact_set dcF by A1,A3;
    end;
    assume
A4: A in charact_set dcF;
    then consider x, y being Subset of X such that
A5: [x, y] in dcF and
A6: x c= AA and
A7: not y c= AA by Th55;
A8: A is Subset of X by A4,Th55;
    assume not A in charact_set F;
    then for x, y being Subset of X st [x, y] in F & x c= AA holds y c= AA
         by A8;
    then
A9: A in B by A8;
    B = enclosure_of F;
    then Dependency-closure F = X deps_encl_by B by Th38;
    then consider a, b being Subset of X such that
A10: [x, y] = [a,b] and
A11: for c being set st c in B & a c= c holds b c= c by A5;
A12: y = b by A10,XTUPLE_0:1;
    x = a by A10,XTUPLE_0:1;
    hence contradiction by A6,A7,A11,A12,A9;
  end;
  hence thesis by TARSKI:2;
end;
