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;

theorem Th36:
  for K1,K2 be maximal SubSimplicialComplex of KX st [#]K1 = [#]K2
    holds the TopStruct of K1 = the TopStruct of K2
 proof
  let K1,K2 be maximal SubSimplicialComplex of KX;
  assume A1: [#]K1=[#]K2;
  set TOP1=the topology of K1,TOP2=the topology of K2,TOP=the topology of KX;
  A2: TOP/\bool[#]K2 c=TOP2 by Th33;
  TOP1 c=TOP by Def13;
  then A3: TOP1 c=TOP/\bool[#]K1 by XBOOLE_1:19;
  TOP2 c=TOP by Def13;
  then A4: TOP2 c=TOP/\bool[#]K2 by XBOOLE_1:19;
  TOP/\bool[#]K1 c=TOP1 by Th33;
  then TOP1=TOP/\bool[#]K1 by A3
   .=TOP2 by A1,A2,A4;
  hence thesis by A1;
 end;
