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<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="21_1_1_1"><span class="kw">now </span></a><div class="add"><span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c2">V1</font> is  <a href="c0sp1.html#V4" title="C0SP1:attr.4">additively-linearly-closed</a>  implies <font color="Maroon" title="c2">V1</font> is  <a href="c0sp1.html#V2" title="C0SP1:attr.2">additively-closed</a>  )</span><br/><span class="kw">assume </span><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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">V1</font> is  <a href="c0sp1.html#V2" title="C0SP1:attr.2">additively-closed</a> </span><br/><a NAME="E2:21_1_1_1"/><span class="kw">then </span>
( <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>  )
 <span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#D10" target="_self" title="C0SP1:def.10">Def10</a></span>;<br/><span class="kw">hence </span><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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/></div><span class="kw">end;</span></div>
<span class="kw">hence </span><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> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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