`∀L,L':(ℕ+ ⟶ ℤ) List.`
`  (((||L|| = ||L'|| ∈ ℤ) ∧ (∀i:ℕ||L||. bdd-diff(L[i];L'[i]))) `` bdd-diff(reg-seq-list-add(L);reg-seq-list-add(L')))`

Proof

Definitions occuring in Statement :  reg-seq-list-add: `reg-seq-list-add(L)` bdd-diff: `bdd-diff(f;g)` select: `L[n]` length: `||as||` list: `T List` int_seg: `{i..j-}` nat_plus: `ℕ+` all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ` equal: `s = t ∈ T`
Definitions unfolded in proof :  all: `∀x:A. B[x]` implies: `P `` Q` and: `P ∧ Q` member: `t ∈ T` squash: `↓T` uall: `∀[x:A]. B[x]` prop: `ℙ` true: `True` subtype_rel: `A ⊆r B` uimplies: `b supposing a` guard: `{T}` iff: `P `⇐⇒` Q` rev_implies: `P `` Q` so_lambda: `λ2x.t[x]` int_seg: `{i..j-}` lelt: `i ≤ j < k` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` not: `¬A` top: `Top` so_apply: `x[s]` map: `map(f;as)` list_ind: list_ind nil: `[]` it: `⋅` select: `L[n]` so_lambda: `λ2x y.t[x; y]` so_apply: `x[s1;s2]` ge: `i ≥ j ` le: `A ≤ B` uiff: `uiff(P;Q)` subtract: `n - m` less_than: `a < b` less_than': `less_than'(a;b)` nat_plus: `ℕ+` cons: `[a / b]` bdd-diff: `bdd-diff(f;g)` nat: `ℕ` rev_uimplies: `rev_uimplies(P;Q)`
Lemmas referenced :  bdd-diff_wf squash_wf true_wf nat_plus_wf reg-seq-list-add-as-l_sum iff_weakening_equal equal_wf length_wf all_wf int_seg_wf select_wf int_seg_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermVar_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_var_lemma int_formula_prop_wf decidable__lt intformless_wf int_formula_prop_less_lemma intformeq_wf int_formula_prop_eq_lemma list_wf list_induction l_sum_wf map_wf equal-wf-base-T nil_wf length_of_nil_lemma stuck-spread base_wf map_nil_lemma l_sum_nil_lemma equal-wf-base length_of_cons_lemma map_cons_lemma l_sum_cons_lemma non_neg_length itermAdd_wf int_term_value_add_lemma cons_wf add-is-int-iff false_wf equal-wf-T-base decidable__equal_int add-member-int_seg2 subtract_wf itermSubtract_wf int_term_value_subtract_lemma lelt_wf select-cons-tl add-subtract-cancel add_nat_plus length_wf_nat less_than_wf nat_plus_properties nat_properties le_wf absval_wf trivial-bdd-diff nat_wf itermMinus_wf int_term_value_minus_lemma and_wf le_functionality le_weakening int-triangle-inequality
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity lambdaFormation sqequalHypSubstitution productElimination thin cut applyEquality lambdaEquality imageElimination introduction extract_by_obid isectElimination hypothesisEquality equalityTransitivity hypothesis equalitySymmetry functionEquality intEquality natural_numberEquality sqequalRule imageMemberEquality baseClosed universeEquality independent_isectElimination independent_functionElimination dependent_functionElimination productEquality because_Cache setElimination rename unionElimination dependent_pairFormation int_eqEquality isect_memberEquality voidElimination voidEquality independent_pairFormation computeAll functionExtensionality addEquality pointwiseFunctionality promote_hyp baseApply closedConclusion dependent_set_memberEquality hyp_replacement applyLambdaEquality

Latex:
\mforall{}L,L':(\mBbbN{}\msupplus{}  {}\mrightarrow{}  \mBbbZ{})  List.
(((||L||  =  ||L'||)  \mwedge{}  (\mforall{}i:\mBbbN{}||L||.  bdd-diff(L[i];L'[i])))