xÎL(G)-{L}                                                   hipótese

SÞ_G^*x                                                            def. L(G)

$k>0 SÞ_G^*S₁^kÞ_G^*x                                       def. P em G (S→S₁S|L únicasproduções com S e x≠L)

$k>0 S₁^kÞ_G^*x                                                def. Þ_G^*

$k>0 S₁^kÞ_G1^*x                                               def. P em G

$k>0 $x₁,…,x_k  x=x_1…x_k e S₁Þ_G1^*x_i(i=1,…,k)  def. Þ_G^*

$k>0 $x₁,…,x_k  x=x_1…x_k e x_iÎL(G₁)  (i=1,…,k) def. L(G)

$k>0 $x_1…x_kx=x_1…x_k e xÎL(G₁) ^k                     def. Lⁿ

xÎL(G₁)*                                                         def. L* e x≠L

R3:

xÎL(G)-{L}                                                  hipótese

SÞ_G^*x                                                             def. Lⁿ

$k>0 SÞ_G^*S₁^kÞ_G^*x                                        def. P em G (S→S₁S|L únicasproduções com S e x≠L)

$k>0 S₁^kÞ_G^*x                                                 def. Þ_G^*

$k>0 S₁^kÞ_G1^*x                                                def. P em G

$k>0 $x₁,…,x_k  x=x_1…x_k e S₁Þ_G1^*x_i(i=1,…,k)  def. Þ_G^*

$k>0 $x₁,…,x_k  x=x_1…x_k e x_iÎL(G₁)  (i=1,…,k) def. L(G)

$k>0 $x_1…x_kx=x_1…x_k e xÎL(G₁) ^k                     def. L(G)

xÎL(G₁)*                                                         def. L* e x≠L

xÎL(G)-{L}                                                     hipótese

SÞ_G^*x                                                            def. Þ_G^*

$k>0 SÞ_G^*S₁^kÞ_G^*x                                       def. P em G (S→S₁S|L únicasproduções com S e x≠L)

$k>0 S₁^kÞ_G^*x                                                def. L(G)

$k>0 S₁^kÞ_G1^*x                                               def. P em G

$k>0 $x₁,…,x_k  x=x_1…x_k e S₁Þ_G1^*x_i(i=1,…,k)  def. Þ_G^*

$k>0 $x₁,…,x_k  x=x_1…x_k e x_iÎL(G₁)  (i=1,…,k) def. L(G)

$k>0 $x_1…x_kx=x_1…x_k e xÎL(G₁) ^k                     def. Lⁿ

xÎL(G₁)*                                                         def. L* e x≠L

ÎL(G)-{L}                                                      hipótese

SÞ_G^*x                                                            def. L(G)

$k>0 SÞ_G^*S₁^kÞ_G^*x                                       def. P em G (S→S₁S|L únicasproduções com S e x≠L)

$k>0 S₁^kÞ_G^*x                                                def. L* e x≠L

$k>0 S₁^kÞ_G1^*x                                               def. P em G

$k>0 $x₁,…,x_k  x=x_1…x_k e S₁Þ_G1^*x_i(i=1,…,k)  def. Þ_G^*

$k>0 $x₁,…,x_k  x=x_1…x_k e x_iÎL(G₁)  (i=1,…,k) def. L(G)

$k>0 $x_1…x_kx=x_1…x_k e xÎL(G₁) ^k                     def. Lⁿ

xÎL(G₁)*                                                         def. Þ_G^*