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<div><span class="kw">theorem </span><a NAME="T22"><span class="comment"><font color="firebrick">:: ZMODUL01:22</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">R</font> being   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Ring</a><br/>  for <font color="Olive" title="b2">V</font> being   <a href="vectsp_2.html#NM5" title="VECTSP_2:NM.5">LeftMod</a> of <font color="Olive" title="b1">R</font><br/>  for <font color="Olive" title="b3">V1</font>, <font color="Olive" title="b4">V2</font>, <font color="Olive" title="b5">V3</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">V</font>  st <font color="Olive" title="b3">V1</font> is  <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a>  &amp; <font color="Olive" title="b4">V2</font> is  <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a>  &amp; <font color="Olive" title="b5">V3</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><font color="Olive" title="b7">v</font> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Olive" title="b6">u</font></span>)</span> where <font color="Olive" title="b6">u</font>, <font color="Olive" title="b7">v</font> is   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Olive" title="b2">V</font> : ( <font color="Olive" title="b7">v</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">V1</font> &amp; <font color="Olive" title="b6">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b4">V2</font> ) </span> } </span>  holds <br/><font color="Olive" title="b5">V3</font> is  <a href="vectsp_4.html#V1" title="VECTSP_4:attr.1">linearly-closed</a> </div></div>
