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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c2">V1</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>;<br/><b>assume </b><a NAME="E1:21_1_1_1"/>
<font color="Maroon" title="c2">V1</font> is <a href="c0sp1.html#V4" title="C0SP1:attr.4">additively-linearly-closed</a>
 ;<br/><a NAME="E2:21_1_1_1"/><b>then </b>
( <font color="Maroon" title="c2">V1</font> is <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> &amp; <font color="Maroon" title="c2">V1</font> is <a href="c0sp1.html#V1" title="C0SP1:attr.1">having-inverse</a> )
 <b>by </b><i><a class="ref" href="c0sp1.html#D10" target="_self" title="C0SP1:def.10">def210</a></i>;<br/><b>hence </b><a NAME="E3:21_1_1_1"/>
<font color="Maroon" title="c2">V1</font> is <a href="c0sp1.html#V2" title="C0SP1:attr.2">additively-closed</a>
 <b>by </b><i><a class="ref" href="c0sp1.html#D2" target="_self" title="C0SP1:def.2">Rdef210</a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:21_1_1"/>
for <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive">b<sub>1</sub></font> is <a href="c0sp1.html#V4" title="C0SP1:attr.4">additively-linearly-closed</a> holds <br/><font color="Olive">b<sub>1</sub></font> is <a href="c0sp1.html#V2" title="C0SP1:attr.2">additively-closed</a>
 ;<br/>


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