src/HOL/Real/HahnBanach/Linearform.thy
 changeset 27611 2c01c0bdb385 parent 25762 c03e9d04b3e4 child 27612 d3eb431db035
```     1.1 --- a/src/HOL/Real/HahnBanach/Linearform.thy	Tue Jul 15 16:02:10 2008 +0200
1.2 +++ b/src/HOL/Real/HahnBanach/Linearform.thy	Tue Jul 15 16:50:09 2008 +0200
1.3 @@ -20,9 +20,10 @@
1.4  declare linearform.intro [intro?]
1.5
1.6  lemma (in linearform) neg [iff]:
1.7 -  includes vectorspace
1.8 +  assumes "vectorspace V"
1.9    shows "x \<in> V \<Longrightarrow> f (- x) = - f x"
1.10  proof -
1.11 +  interpret vectorspace [V] by fact
1.12    assume x: "x \<in> V"
1.13    hence "f (- x) = f ((- 1) \<cdot> x)" by (simp add: negate_eq1)
1.14    also from x have "... = (- 1) * (f x)" by (rule mult)
1.15 @@ -31,9 +32,10 @@
1.16  qed
1.17
1.18  lemma (in linearform) diff [iff]:
1.19 -  includes vectorspace
1.20 +  assumes "vectorspace V"
1.21    shows "x \<in> V \<Longrightarrow> y \<in> V \<Longrightarrow> f (x - y) = f x - f y"
1.22  proof -
1.23 +  interpret vectorspace [V] by fact
1.24    assume x: "x \<in> V" and y: "y \<in> V"
1.25    hence "x - y = x + - y" by (rule diff_eq1)
1.26    also have "f ... = f x + f (- y)" by (rule add) (simp_all add: x y)
1.27 @@ -44,9 +46,10 @@
1.28  text {* Every linear form yields @{text 0} for the @{text 0} vector. *}
1.29
1.30  lemma (in linearform) zero [iff]:
1.31 -  includes vectorspace
1.32 +  assumes "vectorspace V"
1.33    shows "f 0 = 0"
1.34  proof -
1.35 +  interpret vectorspace [V] by fact
1.36    have "f 0 = f (0 - 0)" by simp
1.37    also have "\<dots> = f 0 - f 0" using `vectorspace V` by (rule diff) simp_all
1.38    also have "\<dots> = 0" by simp
```