reserve x,y,X for set,
        r for Real,
        n,k for Nat;

theorem Th3:
  for X be 1-sorted
  for K be subset-closed SimplicialComplexStr of X st K is total
  for S be finite Subset of K st S is simplex-like
    holds Complex_of {@S} is SubSimplicialComplex of K
 proof
  let X be 1-sorted;
  let K be subset-closed SimplicialComplexStr of X such that
   A1: K is total;
  let S be finite Subset of K such that
   A2: S is simplex-like;
  S in the topology of K by A2;
  then A3: {S}c=the topology of K by ZFMISC_1:31;
  set C=Complex_of{@S};
  A4: [#]C c=[#]K by A1;
  the_family_of K is subset-closed;
  then the topology of C c=the topology of K by A3,SIMPLEX0:def 1;
  hence thesis by A4,SIMPLEX0:def 13;
 end;
