`∀[n:ℕ]. ∀[f,g:ℕn ⟶ ℤ].  (increasing(fadd(f;g);n)) supposing (nondecreasing(g;n) and increasing(f;n))`

Proof

Definitions occuring in Statement :  fadd: `fadd(f;g)` nondecreasing: `nondecreasing(f;k)` increasing: `increasing(f;k)` int_seg: `{i..j-}` nat: `ℕ` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` function: `x:A ⟶ B[x]` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  fadd: `fadd(f;g)` increasing: `increasing(f;k)` nondecreasing: `nondecreasing(f;k)` uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` guard: `{T}` nat: `ℕ` int_seg: `{i..j-}` ge: `i ≥ j ` lelt: `i ≤ j < k` and: `P ∧ Q` decidable: `Dec(P)` or: `P ∨ Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` implies: `P `` Q` not: `¬A` top: `Top` prop: `ℙ` le: `A ≤ B` less_than: `a < b` squash: `↓T` uiff: `uiff(P;Q)` subtract: `n - m` so_lambda: `λ2x.t[x]` so_apply: `x[s]`
Lemmas referenced :  nat_wf less_than_wf le_wf all_wf add-member-int_seg2 member-less_than int_seg_wf int_term_value_add_lemma int_formula_prop_le_lemma itermAdd_wf intformle_wf decidable__le lelt_wf int_formula_prop_wf int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_not_lemma int_formula_prop_and_lemma itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf intformnot_wf intformand_wf satisfiable-full-omega-tt decidable__lt nat_properties subtract_wf int_seg_properties
Rules used in proof :  sqequalSubstitution sqequalRule sqequalReflexivity sqequalTransitivity computationStep isect_memberFormation introduction cut lambdaFormation hypothesis sqequalHypSubstitution dependent_functionElimination thin hypothesisEquality because_Cache lemma_by_obid isectElimination natural_numberEquality setElimination rename productElimination addEquality applyEquality dependent_set_memberEquality independent_pairFormation unionElimination independent_isectElimination dependent_pairFormation lambdaEquality int_eqEquality intEquality isect_memberEquality voidElimination voidEquality computeAll imageElimination equalityTransitivity equalitySymmetry functionEquality

Latex:
\mforall{}[n:\mBbbN{}].  \mforall{}[f,g:\mBbbN{}n  {}\mrightarrow{}  \mBbbZ{}].