reserve x,y, X,Y,Z for set,
        D for non empty set,
        n,k for Nat,
        i,i1,i2 for Integer;
reserve K for SimplicialComplexStr;
reserve KX for SimplicialComplexStr of X,
        SX for SubSimplicialComplex of KX;
reserve SC for SimplicialComplex of X;

theorem
 for i being dim-like number
   st KX is subset-closed & Skeleton_of(KX,i) is empty-membered
   holds KX is empty-membered or i = -1
 proof let i be dim-like number;
  assume KX is subset-closed;
  then A1: the_family_of KX is subset-closed;
  assume A2: Skeleton_of(KX,i) is empty-membered;
  assume KX is with_non-empty_element;
  then the topology of KX is with_non-empty_element;
  then consider x be non empty set such that
   A3: x in the topology of KX;
  consider y being object such that
   A4: y in x by XBOOLE_0:def 1;
  assume i<>-1;
  then {}c=card Segm(i+1) & Segm(i+1) is non empty;
  then {} in card Segm(i+1) by CARD_1:3;
  then 1 c=card Segm (i+1) by CARD_2:68;
  then A5: card{y}c=card Segm(i+1) by CARD_1:30;
  {y}c=x by A4,ZFMISC_1:31;
  then {y} in the topology of KX by A1,A3;
  then {y} in the_subsets_with_limited_card(Segm(i+1),the topology of KX)
  by A5,Def2;
  then A6: {y} in the topology of Skeleton_of(KX,i) by Th2;
  the topology of Skeleton_of(KX,i) is empty-membered by A2;
  hence thesis by A6;
 end;
