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<div><span class="kw">theorem </span><a NAME="T31"><span class="comment"><font color="firebrick">:: ZMODUL01:31</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">u</font>, <font color="Olive" title="b4">v</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Olive" title="b2">V</font><br/>  for <font color="Olive" title="b5">W</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b2">V</font><br/>  for <font color="Olive" title="b6">w1</font>, <font color="Olive" title="b7">w2</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Olive" title="b5">W</font>  st <font color="Olive" title="b6">w1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v</font> &amp; <font color="Olive" title="b7">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">u</font> holds <br/><font color="Olive" title="b6">w1</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b7">w2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">v</font> <a href="algstr_0.html#K5" title="ALGSTR_0:func.5">-</a> <font color="Olive" title="b3">u</font> <span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_4.html#T16" onmouseover="rs('vectsp_4/T16')" onmouseout="rh()">VECTSP_4:16</a></span>;<br/></div></div>
