prime_valuation_distinct_prime_power_zero

forall p q k z. (~((p) = 1) /\ forall pvs_left_distinct_base pvs_right_distinct_base. (p) = pvs_left_distinct_base * pvs_right_distinct_base -> pvs_left_distinct_base = 1 \/ pvs_right_distinct_base = 1) -> (~((q) = 1) /\ forall pvs_left_distinct_valuation pvs_right_distinct_valuation. (q) = pvs_left_distinct_valuation * pvs_right_distinct_valuation -> pvs_left_distinct_valuation = 1 \/ pvs_right_distinct_valuation = 1) -> ~(q = p) -> (exists pa_b_pvs_distinct_power pa_c_pvs_distinct_power. ((forall pa_i_pvs_distinct_power_repeat. (exists pa_lt_pvs_distinct_power_repeat_bound. pa_lt_pvs_distinct_power_repeat_bound + S pa_i_pvs_distinct_power_repeat = k) -> (((exists pa_h_pvs_distinct_power_repeat_decoded. pa_h_pvs_distinct_power_repeat_decoded + S (p) = S ((S (pa_i_pvs_distinct_power_repeat)) * pa_c_pvs_distinct_power)) /\ exists pa_q_pvs_distinct_power_repeat_decoded. pa_b_pvs_distinct_power = pa_q_pvs_distinct_power_repeat_decoded * S ((S (pa_i_pvs_distinct_power_repeat)) * pa_c_pvs_distinct_power) + (p)))) /\ (exists pa_u_pvs_distinct_power_product pa_v_pvs_distinct_power_product. ((((exists pa_h_pvs_distinct_power_product_start. pa_h_pvs_distinct_power_product_start + S (1) = S ((S (0)) * pa_v_pvs_distinct_power_product)) /\ exists pa_q_pvs_distinct_power_product_start. pa_u_pvs_distinct_power_product = pa_q_pvs_distinct_power_product_start * S ((S (0)) * pa_v_pvs_distinct_power_product) + (1))) /\ ((((exists pa_h_pvs_distinct_power_product_terminal. pa_h_pvs_distinct_power_product_terminal + S (z) = S ((S (k)) * pa_v_pvs_distinct_power_product)) /\ exists pa_q_pvs_distinct_power_product_terminal. pa_u_pvs_distinct_power_product = pa_q_pvs_distinct_power_product_terminal * S ((S (k)) * pa_v_pvs_distinct_power_product) + (z))) /\ forall pa_i_pvs_distinct_power_product. (exists pa_lt_pvs_distinct_power_product_bound. pa_lt_pvs_distinct_power_product_bound + S pa_i_pvs_distinct_power_product = k) -> exists pa_p_pvs_distinct_power_product pa_r_pvs_distinct_power_product pa_s_pvs_distinct_power_product. ((((exists pa_h_pvs_distinct_power_product_factor. pa_h_pvs_distinct_power_product_factor + S (pa_p_pvs_distinct_power_product) = S ((S (pa_i_pvs_distinct_power_product)) * pa_c_pvs_distinct_power)) /\ exists pa_q_pvs_distinct_power_product_factor. pa_b_pvs_distinct_power = pa_q_pvs_distinct_power_product_factor * S ((S (pa_i_pvs_distinct_power_product)) * pa_c_pvs_distinct_power) + (pa_p_pvs_distinct_power_product))) /\ ((((exists pa_h_pvs_distinct_power_product_partial. pa_h_pvs_distinct_power_product_partial + S (pa_r_pvs_distinct_power_product) = S ((S (pa_i_pvs_distinct_power_product)) * pa_v_pvs_distinct_power_product)) /\ exists pa_q_pvs_distinct_power_product_partial. pa_u_pvs_distinct_power_product = pa_q_pvs_distinct_power_product_partial * S ((S (pa_i_pvs_distinct_power_product)) * pa_v_pvs_distinct_power_product) + (pa_r_pvs_distinct_power_product))) /\ ((((exists pa_h_pvs_distinct_power_product_successor. pa_h_pvs_distinct_power_product_successor + S (pa_s_pvs_distinct_power_product) = S ((S (S pa_i_pvs_distinct_power_product)) * pa_v_pvs_distinct_power_product)) /\ exists pa_q_pvs_distinct_power_product_successor. pa_u_pvs_distinct_power_product = pa_q_pvs_distinct_power_product_successor * S ((S (S pa_i_pvs_distinct_power_product)) * pa_v_pvs_distinct_power_product) + (pa_s_pvs_distinct_power_product))) /\ pa_s_pvs_distinct_power_product = pa_r_pvs_distinct_power_product * pa_p_pvs_distinct_power_product)))))))) -> (((exists bpd_gap_pvs_distinct_zero_selected_bound. bpd_gap_pvs_distinct_zero_selected_bound + (0) = (z)) /\ (exists bpvi_result_pvs_distinct_zero_selected. ((exists bpvi_b_pvs_distinct_zero_selected_power bpvi_c_pvs_distinct_zero_selected_power. ((forall bpvi_i_pvs_distinct_zero_selected_power. (exists bpvi_repeat_gap_pvs_distinct_zero_selected_power. bpvi_repeat_gap_pvs_distinct_zero_selected_power + S bpvi_i_pvs_distinct_zero_selected_power = 0) -> (((exists bpvi_h_pvs_distinct_zero_selected_power_repeat. bpvi_h_pvs_distinct_zero_selected_power_repeat + S (q) = S ((S (bpvi_i_pvs_distinct_zero_selected_power)) * bpvi_c_pvs_distinct_zero_selected_power)) /\ exists bpvi_q_pvs_distinct_zero_selected_power_repeat. bpvi_b_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_repeat * S ((S (bpvi_i_pvs_distinct_zero_selected_power)) * bpvi_c_pvs_distinct_zero_selected_power) + (q)))) /\ (exists bpvi_u_pvs_distinct_zero_selected_power bpvi_v_pvs_distinct_zero_selected_power. ((((exists bpvi_h_pvs_distinct_zero_selected_power_start. bpvi_h_pvs_distinct_zero_selected_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_distinct_zero_selected_power)) /\ exists bpvi_q_pvs_distinct_zero_selected_power_start. bpvi_u_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_start * S ((S (0)) * bpvi_v_pvs_distinct_zero_selected_power) + (1))) /\ ((((exists bpvi_h_pvs_distinct_zero_selected_power_terminal. bpvi_h_pvs_distinct_zero_selected_power_terminal + S (bpvi_result_pvs_distinct_zero_selected) = S ((S (0)) * bpvi_v_pvs_distinct_zero_selected_power)) /\ exists bpvi_q_pvs_distinct_zero_selected_power_terminal. bpvi_u_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_terminal * S ((S (0)) * bpvi_v_pvs_distinct_zero_selected_power) + (bpvi_result_pvs_distinct_zero_selected))) /\ forall bpvi_j_pvs_distinct_zero_selected_power. (exists bpvi_product_gap_pvs_distinct_zero_selected_power. bpvi_product_gap_pvs_distinct_zero_selected_power + S bpvi_j_pvs_distinct_zero_selected_power = 0) -> exists bpvi_factor_pvs_distinct_zero_selected_power bpvi_partial_pvs_distinct_zero_selected_power bpvi_successor_pvs_distinct_zero_selected_power. ((((exists bpvi_h_pvs_distinct_zero_selected_power_factor. bpvi_h_pvs_distinct_zero_selected_power_factor + S (bpvi_factor_pvs_distinct_zero_selected_power) = S ((S (bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_c_pvs_distinct_zero_selected_power)) /\ exists bpvi_q_pvs_distinct_zero_selected_power_factor. bpvi_b_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_factor * S ((S (bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_c_pvs_distinct_zero_selected_power) + (bpvi_factor_pvs_distinct_zero_selected_power))) /\ ((((exists bpvi_h_pvs_distinct_zero_selected_power_partial. bpvi_h_pvs_distinct_zero_selected_power_partial + S (bpvi_partial_pvs_distinct_zero_selected_power) = S ((S (bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_v_pvs_distinct_zero_selected_power)) /\ exists bpvi_q_pvs_distinct_zero_selected_power_partial. bpvi_u_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_partial * S ((S (bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_v_pvs_distinct_zero_selected_power) + (bpvi_partial_pvs_distinct_zero_selected_power))) /\ ((((exists bpvi_h_pvs_distinct_zero_selected_power_successor. bpvi_h_pvs_distinct_zero_selected_power_successor + S (bpvi_successor_pvs_distinct_zero_selected_power) = S ((S (S bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_v_pvs_distinct_zero_selected_power)) /\ exists bpvi_q_pvs_distinct_zero_selected_power_successor. bpvi_u_pvs_distinct_zero_selected_power = bpvi_q_pvs_distinct_zero_selected_power_successor * S ((S (S bpvi_j_pvs_distinct_zero_selected_power)) * bpvi_v_pvs_distinct_zero_selected_power) + (bpvi_successor_pvs_distinct_zero_selected_power))) /\ bpvi_successor_pvs_distinct_zero_selected_power = bpvi_partial_pvs_distinct_zero_selected_power * bpvi_factor_pvs_distinct_zero_selected_power)))))))) /\ exists bpvi_divisor_factor_pvs_distinct_zero_selected. z = bpvi_result_pvs_distinct_zero_selected * bpvi_divisor_factor_pvs_distinct_zero_selected))) /\ forall bpd_candidate_pvs_distinct_zero. (exists bpd_gap_pvs_distinct_zero_candidate_bound. bpd_gap_pvs_distinct_zero_candidate_bound + (bpd_candidate_pvs_distinct_zero) = (z)) -> (exists bpvi_result_pvs_distinct_zero_candidate. ((exists bpvi_b_pvs_distinct_zero_candidate_power bpvi_c_pvs_distinct_zero_candidate_power. ((forall bpvi_i_pvs_distinct_zero_candidate_power. (exists bpvi_repeat_gap_pvs_distinct_zero_candidate_power. bpvi_repeat_gap_pvs_distinct_zero_candidate_power + S bpvi_i_pvs_distinct_zero_candidate_power = bpd_candidate_pvs_distinct_zero) -> (((exists bpvi_h_pvs_distinct_zero_candidate_power_repeat. bpvi_h_pvs_distinct_zero_candidate_power_repeat + S (q) = S ((S (bpvi_i_pvs_distinct_zero_candidate_power)) * bpvi_c_pvs_distinct_zero_candidate_power)) /\ exists bpvi_q_pvs_distinct_zero_candidate_power_repeat. bpvi_b_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_repeat * S ((S (bpvi_i_pvs_distinct_zero_candidate_power)) * bpvi_c_pvs_distinct_zero_candidate_power) + (q)))) /\ (exists bpvi_u_pvs_distinct_zero_candidate_power bpvi_v_pvs_distinct_zero_candidate_power. ((((exists bpvi_h_pvs_distinct_zero_candidate_power_start. bpvi_h_pvs_distinct_zero_candidate_power_start + S (1) = S ((S (0)) * bpvi_v_pvs_distinct_zero_candidate_power)) /\ exists bpvi_q_pvs_distinct_zero_candidate_power_start. bpvi_u_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_start * S ((S (0)) * bpvi_v_pvs_distinct_zero_candidate_power) + (1))) /\ ((((exists bpvi_h_pvs_distinct_zero_candidate_power_terminal. bpvi_h_pvs_distinct_zero_candidate_power_terminal + S (bpvi_result_pvs_distinct_zero_candidate) = S ((S (bpd_candidate_pvs_distinct_zero)) * bpvi_v_pvs_distinct_zero_candidate_power)) /\ exists bpvi_q_pvs_distinct_zero_candidate_power_terminal. bpvi_u_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_terminal * S ((S (bpd_candidate_pvs_distinct_zero)) * bpvi_v_pvs_distinct_zero_candidate_power) + (bpvi_result_pvs_distinct_zero_candidate))) /\ forall bpvi_j_pvs_distinct_zero_candidate_power. (exists bpvi_product_gap_pvs_distinct_zero_candidate_power. bpvi_product_gap_pvs_distinct_zero_candidate_power + S bpvi_j_pvs_distinct_zero_candidate_power = bpd_candidate_pvs_distinct_zero) -> exists bpvi_factor_pvs_distinct_zero_candidate_power bpvi_partial_pvs_distinct_zero_candidate_power bpvi_successor_pvs_distinct_zero_candidate_power. ((((exists bpvi_h_pvs_distinct_zero_candidate_power_factor. bpvi_h_pvs_distinct_zero_candidate_power_factor + S (bpvi_factor_pvs_distinct_zero_candidate_power) = S ((S (bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_c_pvs_distinct_zero_candidate_power)) /\ exists bpvi_q_pvs_distinct_zero_candidate_power_factor. bpvi_b_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_factor * S ((S (bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_c_pvs_distinct_zero_candidate_power) + (bpvi_factor_pvs_distinct_zero_candidate_power))) /\ ((((exists bpvi_h_pvs_distinct_zero_candidate_power_partial. bpvi_h_pvs_distinct_zero_candidate_power_partial + S (bpvi_partial_pvs_distinct_zero_candidate_power) = S ((S (bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_v_pvs_distinct_zero_candidate_power)) /\ exists bpvi_q_pvs_distinct_zero_candidate_power_partial. bpvi_u_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_partial * S ((S (bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_v_pvs_distinct_zero_candidate_power) + (bpvi_partial_pvs_distinct_zero_candidate_power))) /\ ((((exists bpvi_h_pvs_distinct_zero_candidate_power_successor. bpvi_h_pvs_distinct_zero_candidate_power_successor + S (bpvi_successor_pvs_distinct_zero_candidate_power) = S ((S (S bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_v_pvs_distinct_zero_candidate_power)) /\ exists bpvi_q_pvs_distinct_zero_candidate_power_successor. bpvi_u_pvs_distinct_zero_candidate_power = bpvi_q_pvs_distinct_zero_candidate_power_successor * S ((S (S bpvi_j_pvs_distinct_zero_candidate_power)) * bpvi_v_pvs_distinct_zero_candidate_power) + (bpvi_successor_pvs_distinct_zero_candidate_power))) /\ bpvi_successor_pvs_distinct_zero_candidate_power = bpvi_partial_pvs_distinct_zero_candidate_power * bpvi_factor_pvs_distinct_zero_candidate_power)))))))) /\ exists bpvi_divisor_factor_pvs_distinct_zero_candidate. z = bpvi_result_pvs_distinct_zero_candidate * bpvi_divisor_factor_pvs_distinct_zero_candidate)) -> (exists bpd_gap_pvs_distinct_zero_maximal. bpd_gap_pvs_distinct_zero_maximal + (bpd_candidate_pvs_distinct_zero) = (0)))