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;
reserve K for non void subset-closed SimplicialComplexStr;

theorem
  for K be non void subset-closed SimplicialComplexStr of D
    for S be non void SubSimplicialComplex of K
      for i be Integer,A be Simplex of i,S st
          A is non empty or i <= degree S or degree S = degree K
        holds A is Simplex of i,K
  proof
  let K be non void subset-closed SimplicialComplexStr of D;
  let S be non void SubSimplicialComplex of K;
  let i be Integer,A be Simplex of i,S such that
   A1: A is non empty or i<=degree S or degree S=degree K;
  A in the topology of S & the topology of S c=the topology of K by Def13,
PRE_TOPC:def 2;
  then A in the topology of K;
  then reconsider B=A as Simplex of K by PRE_TOPC:def 2;
  A2: degree S<=degree K by Th32;
  per cases by A1;
  suppose A3: A is non empty;
   then A4: -1<=i by Def18;
   A5: i<=degree S by A3,Def18;
   -1<=i by A3,Def18;
   then A6: card B=i+1 by A5,Def18;
   i<=degree K by A2,A5,XXREAL_0:2;
   hence thesis by A6,A4,Def18;
  end;
  suppose A7: i<=degree S;
   then A8: i<=degree K by A2,XXREAL_0:2;
   per cases;
   suppose A9: -1<=i;
    then card B=i+1 by A7,Def18;
    hence thesis by A8,A9,Def18;
   end;
   suppose A10: -1>i;
    then B is empty by Def18;
    hence thesis by A10,Def18;
   end;
  end;
  suppose A11: degree S=degree K;
   per cases;
   suppose A12: -1<=i & i<=degree S;
    then card B=i+1 by Def18;
    hence thesis by A11,A12,Def18;
   end;
   suppose A13: -1>i or i>degree S;
    then B is empty by Def18;
    hence thesis by A11,A13,Def18;
   end;
  end;
 end;
