### Nuprl Lemma : rsum-split-last

`∀[n,m:ℤ]. ∀[x:{n..m + 1-} ⟶ ℝ].  Σ{x[i] | n≤i≤m} = (Σ{x[i] | n≤i≤m - 1} + x[m]) supposing n ≤ m`

Proof

Definitions occuring in Statement :  rsum: `Σ{x[k] | n≤k≤m}` req: `x = y` radd: `a + b` real: `ℝ` int_seg: `{i..j-}` uimplies: `b supposing a` uall: `∀[x:A]. B[x]` so_apply: `x[s]` le: `A ≤ B` function: `x:A ⟶ B[x]` subtract: `n - m` add: `n + m` natural_number: `\$n` int: `ℤ`
Definitions unfolded in proof :  uall: `∀[x:A]. B[x]` member: `t ∈ T` uimplies: `b supposing a` all: `∀x:A. B[x]` decidable: `Dec(P)` or: `P ∨ Q` so_lambda: `λ2x.t[x]` so_apply: `x[s]` int_seg: `{i..j-}` lelt: `i ≤ j < k` and: `P ∧ Q` less_than: `a < b` squash: `↓T` not: `¬A` implies: `P `` Q` satisfiable_int_formula: `satisfiable_int_formula(fmla)` exists: `∃x:A. B[x]` false: `False` prop: `ℙ` subtract: `n - m` uiff: `uiff(P;Q)` top: `Top` sq_type: `SQType(T)` guard: `{T}` subtype_rel: `A ⊆r B` rev_uimplies: `rev_uimplies(P;Q)` itermAdd: `left (+) right` itermVar: `vvar` itermSubtract: `left (-) right` int_term_ind: int_term_ind real_term_value: `real_term_value(f;t)` req_int_terms: `t1 ≡ t2` itermConstant: `"const"`

Latex:
\mforall{}[n,m:\mBbbZ{}].  \mforall{}[x:\{n..m  +  1\msupminus{}\}  {}\mrightarrow{}  \mBbbR{}].    \mSigma{}\{x[i]  |  n\mleq{}i\mleq{}m\}  =  (\mSigma{}\{x[i]  |  n\mleq{}i\mleq{}m  -  1\}  +  x[m])  supposing  n  \mleq{}  m

Date html generated: 2020_05_20-AM-11_11_33
Last ObjectModification: 2019_12_14-PM-00_56_59

Theory : reals

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