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 Th37:
  for A be Subset of SC holds
    the topology of SC|A = bool A /\ the topology of SC
 proof
  let A be Subset of SC;
  A1: [#](SC|A)=A by Def16;
  then A2: bool A/\the topology of SC c=the topology of SC|A by Th33;
  the topology of SC|A c=the topology of SC by Def13;
  then the topology of SC|A c=bool A/\the topology of SC by A1,XBOOLE_1:19;
  hence thesis by A2;
 end;
