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;

theorem Th14:
  for S be Subset-Family of K holds
    S is simplex-like iff S c= the topology of K
  proof
  let S be Subset-Family of K;
  hereby assume A1: S is simplex-like;
   thus S c=the topology of K
   proof
    let x be object;
    assume A2: x in S;
    then reconsider A=x as Subset of K;
    A is simplex-like by A1,A2;
    hence thesis;
   end;
  end;
  assume A3: S c=the topology of K;
  let A be Subset of K;
  assume A in S;
  hence thesis by A3;
 end;
