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<span class="kw">reconsider </span><font color="Maroon" title="c2">W</font> =  <a href="struct_0.html#K2" title="STRUCT_0:func.2">[#]</a> <font color="Maroon" title="c1">V</font> as   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> ;<br/>
<a NAME="E2:9_1_1"/>
( <font color="Maroon" title="c2">W</font> is  <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a>  &amp; <font color="Maroon" title="c2">W</font> is  <a href="c0sp1.html#V1" title="C0SP1:attr.1">having-inverse</a>  &amp; (  for <font color="Olive" title="b1">a</font> being   <a href="cfunct_1.html#NM1" title="CFUNCT_1:NM.1">Complex</a><br/>  for <font color="Olive" title="b2">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> holds <br/><font color="Olive" title="b1">a</font> <a href="clvect_1.html#K1" title="CLVECT_1:func.1">*</a> <font color="Olive" title="b2">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> ) )
 
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<a NAME="E3:9_1_1"/><span class="kw">then </span><span class="lab"><font color="Green" title="E5">A1</font></span>: 
<font color="Maroon" title="c2">W</font> is  <a href="cc0sp1.html#V1" title="CC0SP1:attr.1">Cadditively-linearly-closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="cc0sp1.html#D2" target="_self" title="CC0SP1:def.2">Def10</a></span>;<br/>
<a NAME="E4:9_1_1"/>
(  <a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">V</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> &amp; (  for <font color="Olive" title="b1">v</font>, <font color="Olive" title="b2">u</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">V</font>  st <font color="Olive" title="b1">v</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> &amp; <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> holds <br/><font color="Olive" title="b1">v</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">W</font> ) )
 
;<br/>
<a NAME="E5:9_1_1"/><span class="kw">then </span>
<font color="Maroon" title="c2">W</font> is  <a href="c0sp1.html#V3" title="C0SP1:attr.3">multiplicatively-closed</a> 
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="c0sp1.html#D4" title="C0SP1:def.4">C0SP1:def 4</a></span>;<br/>
<span class="kw">hence </span><a NAME="E6:9_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">V</font> st <br/>( <font color="Olive">b<sub>1</sub></font> is  <a href="cc0sp1.html#V1" title="CC0SP1:attr.1">Cadditively-linearly-closed</a>  &amp; <font color="Olive">b<sub>1</sub></font> is  <a href="c0sp1.html#V3" title="C0SP1:attr.3">multiplicatively-closed</a>  )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E3:9_1_1"><span class="lab"><font color="Green" title="E5">A1</font></span></a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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