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 i1,i2 being dim-like number st -1 <= i1 & i1 <= i2
  holds Skeleton_of(KX,i1) is SubSimplicialComplex of Skeleton_of(KX,i2)
 proof let i1,i2 be dim-like number;
  assume that
    -1<=i1 and
   A1: i1<=i2;
  reconsider I1=i1+1,I2=i2+1 as Element of NAT by INT_1:3;
  the_subsets_with_limited_card(Segm(i1+1),the topology of KX)c=
the_subsets_with_limited_card(Segm(i2+1),the topology of KX)
  proof
   let x be object;
    reconsider xx=x as set by TARSKI:1;
   I1<=I2 by A1,XREAL_1:6;
   then A2: card Segm I1 c=card Segm I2 by NAT_1:40;
   assume
A3: x in the_subsets_with_limited_card(Segm(i1+1),the topology of KX);
   then card xx c=card I1 by Def2;
   then A4: card xx c=card I2 by A2;
   x in the topology of KX by A3,Def2;
   hence thesis by A4,Def2;
  end;
  then the_subsets_with_limited_card(Segm(i1+1),the topology of KX)
  is_finer_than
  the_subsets_with_limited_card(Segm(i2+1),the topology of KX);
  then A5: the topology of Skeleton_of(KX,i1)c=the topology of Skeleton_of(KX,
i2) by Th6;
  [#]Skeleton_of(KX,i1)=[#]Skeleton_of(KX,i2);
  hence thesis by A5,Def13;
 end;
