`∀[r:CRng]. ∀[f,g:PowerSeries(r)]. ∀[y:Atom]. ∀[n:ℕ].  ([(f+g)]_n(y:=1) = ([f]_n(y:=1)+[g]_n(y:=1)) ∈ PowerSeries(r))`

Proof

Definitions occuring in Statement :  fps-set-to-one: `[f]_n(y:=1)` fps-add: `(f+g)` power-series: `PowerSeries(X;r)` nat: `ℕ` uall: `∀[x:A]. B[x]` atom: `Atom` equal: `s = t ∈ T` crng: `CRng`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uiff: `uiff(P;Q)` and: `P ∧ Q` uimplies: `b supposing a` all: `∀x:A. B[x]` fps-set-to-one: `[f]_n(y:=1)` fps-add: `(f+g)` fps-coeff: `f[b]` subtype_rel: `A ⊆r B` implies: `P `` Q` bool: `𝔹` unit: `Unit` it: `⋅` btrue: `tt` nat: `ℕ` bor: `p ∨bq` ifthenelse: `if b then t else f fi ` bfalse: `ff` exists: `∃x:A. B[x]` prop: `ℙ` or: `P ∨ Q` sq_type: `SQType(T)` guard: `{T}` bnot: `¬bb` assert: `↑b` false: `False` not: `¬A` crng: `CRng` rng: `Rng` power-series: `PowerSeries(X;r)` ge: `i ≥ j ` decidable: `Dec(P)` satisfiable_int_formula: `satisfiable_int_formula(fmla)` top: `Top` true: `True` squash: `↓T` infix_ap: `x f y` iff: `P `⇐⇒` Q` rev_implies: `P `` Q`
Lemmas referenced :  fps-ext fps-set-to-one_wf fps-add_wf lt_int_wf bag-count_wf atom-deq_wf bool_wf eqtt_to_assert assert_of_lt_int nat_wf eqff_to_assert equal_wf bool_cases_sqequal subtype_base_sq bool_subtype_base assert-bnot less_than_wf bag-size_wf rng_plus_wf bag-append_wf bag-rep_wf subtract_wf nat_properties decidable__le satisfiable-full-omega-tt intformand_wf intformnot_wf intformle_wf itermConstant_wf itermSubtract_wf itermVar_wf intformless_wf int_formula_prop_and_lemma int_formula_prop_not_lemma int_formula_prop_le_lemma int_term_value_constant_lemma int_term_value_subtract_lemma int_term_value_var_lemma int_formula_prop_less_lemma int_formula_prop_wf le_wf list-subtype-bag subtype_rel_self bag_wf power-series_wf crng_wf rng_car_wf rng_zero_wf rng_plus_zero iff_weakening_equal
Rules used in proof :  sqequalSubstitution sqequalTransitivity computationStep sqequalReflexivity isect_memberFormation introduction cut extract_by_obid sqequalHypSubstitution isectElimination thin because_Cache hypothesisEquality atomEquality hypothesis productElimination independent_isectElimination lambdaFormation sqequalRule natural_numberEquality applyEquality unionElimination equalityElimination equalityTransitivity equalitySymmetry lambdaEquality setElimination rename dependent_pairFormation promote_hyp dependent_functionElimination instantiate cumulativity independent_functionElimination voidElimination dependent_set_memberEquality int_eqEquality intEquality isect_memberEquality voidEquality independent_pairFormation computeAll axiomEquality imageElimination imageMemberEquality baseClosed

Latex:
\mforall{}[r:CRng].  \mforall{}[f,g:PowerSeries(r)].  \mforall{}[y:Atom].  \mforall{}[n:\mBbbN{}].    ([(f+g)]\_n(y:=1)  =  ([f]\_n(y:=1)+[g]\_n(y:=1)))

Date html generated: 2018_05_21-PM-10_12_51
Last ObjectModification: 2017_07_26-PM-06_35_10

Theory : power!series

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