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 Th33:
  SX is maximal iff bool[#]SX /\ the topology of KX c= the topology of SX
 proof
  hereby assume A1: SX is maximal;
   thus bool[#]SX/\the topology of KX c=the topology of SX
   proof
    let x be object;
    assume A2: x in bool[#]SX/\the topology of KX;
    then reconsider A=x as Subset of SX by XBOOLE_0:def 4;
    A in the topology of KX by A2,XBOOLE_0:def 4;
    then A is simplex-like by A1;
    hence thesis;
   end;
  end;
  assume A3: bool[#]SX/\the topology of KX c=the topology of SX;
  let A be Subset of SX;
  assume A in the topology of KX;
  then A in bool[#]SX/\the topology of KX by XBOOLE_0:def 4;
  hence thesis by A3;
 end;
