reserve x,y,X for set,
        r for Real,
        n,k for Nat;
reserve RLS for non empty RLSStruct,
        Kr,K1r,K2r for SimplicialComplexStr of RLS,
        V for RealLinearSpace,
        Kv for non void SimplicialComplex of V;

theorem
  for A,B be Subset-Family of RLS holds
    |.Complex_of (A\/B).| = |.Complex_of A.| \/ |.Complex_of B.|
  proof
  let A,B be Subset-Family of RLS;
  set CA=Complex_of A,CB=Complex_of B,CAB=Complex_of(A\/B);
  A1: (the topology of CA)\/the topology of CB=the topology of CAB by
SIMPLEX0:5;
  thus|.CAB.|c=|.CA.|\/|.CB.|
  proof
   let x be object;
   assume x in |.CAB.|;
   then consider S be Subset of CAB such that
    A2: S is simplex-like and
    A3: x in conv@S by Def3;
   A4: S in the topology of CAB by A2;
   per cases by A1,A4,XBOOLE_0:def 3;
   suppose A5: S in the topology of CA;
    reconsider S1=S as Subset of CA;
    @S=@S1 & S1 is simplex-like by A5;
    then conv@S c=|.CA.| by Th5;
    hence thesis by A3,XBOOLE_0:def 3;
   end;
   suppose A6: S in the topology of CB;
    reconsider S1=S as Subset of CB;
    @S=@S1 & S1 is simplex-like by A6;
    then conv@S c=|.CB.| by Th5;
    hence thesis by A3,XBOOLE_0:def 3;
   end;
  end;
  |.CA.|c=|.CAB.| & |.CB.|c=|.CAB.| by A1,Th4,XBOOLE_1:7;
  hence thesis by XBOOLE_1:8;
 end;
