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 A,B be Subset of SC for B1 be Subset of SC|A st B1 = B
   holds SC|A|B1 = SC|B
 proof
  let A,B be Subset of SC;
  let B1 be Subset of SC|A;
  reconsider SCAB=SC|A|B1 as maximal strict SubSimplicialComplex of SC by Th34;
  assume A1: B1=B;
  [#]SCAB=B1 by Def16;
  hence thesis by A1,Def16;
 end;
