reserve            x for object,
               X,Y,Z for set,
         i,j,k,l,m,n for Nat,
                 r,s for Real,
                  no for Element of OrderedNAT,
                   A for Subset of [:NAT,NAT:];
reserve X,Y,X1,X2 for non empty set,
          cA1,cB1 for filter_base of X1,
          cA2,cB2 for filter_base of X2,
              cF1 for Filter of X1,
              cF2 for Filter of X2,
             cBa1 for basis of cF1,
             cBa2 for basis of cF2;

theorem Th22:
  cF1 = <. cB1.) & cF1 = <. cA1.) &
  cF2 = <. cB2.) & cF2 = <. cA2.) implies
  <. [:cB1,cB2:] .) = <. [:cA1,cA2:] .)
  proof
    assume that
A1: cF1 = <. cB1.) and
A2: cF1 = <. cA1.) and
A3: cF2 = <. cB2.) and
A4: cF2 = <. cA2.);
A5: cB1,cA1 are_equivalent_generators by A1,A2,CARDFIL2:26;
A6: cB2,cA2 are_equivalent_generators by A3,A4,CARDFIL2:26;
    now
      hereby
        let B be Element of [:cB1,cB2:];
        B in [:cB1,cB2:];
        then consider B1 be Element of cB1, B2 be Element of cB2 such that
A7:     B = [:B1,B2:];
        consider B1A1 be Element of cA1 such that
A8:     B1A1 c= B1 by A5;
        consider B2A2 be Element of cA2 such that
A9:     B2A2 c= B2 by A6;
        [:B1A1,B2A2:] in [:cA1,cA2:];
        then reconsider A = [:B1A1,B2A2:] as Element of [:cA1,cA2:];
        [:B1A1,B2A2:] c= [:B1,B2A2:] & [:B1,B2A2:] c= [:B1,B2:]
          by A8,A9,ZFMISC_1:95;
        then A c= B by A7;
        hence ex A be Element of [:cA1,cA2:] st A c= B;
      end;
      hereby
        let A be Element of [:cA1,cA2:];
        A in [:cA1,cA2:];
        then consider A1 be Element of cA1, A2 be Element of cA2 such that
A10:    A = [:A1,A2:];
        consider A1B1 be Element of cB1 such that
A11:    A1B1 c= A1 by A5;
        consider A2B2 be Element of cB2 such that
A12:    A2B2 c= A2 by A6;
        [:A1B1,A2B2:] in [:cB1,cB2:];
        then reconsider B = [:A1B1,A2B2:] as Element of [:cB1,cB2:];
        [:A1B1,A2B2:] c= [:A1,A2B2:] & [:A1,A2B2:] c= [:A1,A2:]
          by A11,A12,ZFMISC_1:95;
        then B c= A by A10;
        hence ex B be Element of [:cB1,cB2:] st B c= A;
      end;
    end;
    then [:cB1,cB2:], [:cA1,cA2:] are_equivalent_generators;
    hence thesis by CARDFIL2:20;
  end;
