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<div><span class="kw">theorem </span><a NAME="T7"><span class="comment"><font color="firebrick">:: ZMODUL05:7</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">V</font> being   <a href="vectsp_7.html#V3" title="VECTSP_7:attr.3">free</a>   <a href="zmodul03.html#V1" title="ZMODUL03:attr.1">finite-rank</a>  <a href="zmodul01.html#NM1" title="ZMODUL01:NM.1">Z_Module</a><br/>  for <font color="Olive" title="b2">W</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b1">V</font><br/>  for <font color="Olive" title="b3">n</font> being   <a href="ordinal1.html#NM6" title="ORDINAL1:NM.6">Nat</a> holds <br/> ( <font color="Olive" title="b3">n</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Olive" title="b1">V</font> iff  ex <font color="Olive" title="b4">W</font> being   <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">strict</a>   <a href="vectsp_7.html#V3" title="VECTSP_7:attr.3">free</a>  <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b1">V</font> st  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Olive" title="b4">W</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">n</font> ) <span class="kw">by</span> <span class="lab"><a class="txt" href="zmodul05.html#E7"><span class="lab"><font color="Green" title="E4">RL5Lm2</font></span></a>, <a class="ref" href="zmodul05.html#T2" target="_self" onmouseover="rs('zmodul05/T2')" onmouseout="rh()">RL5Th29</a></span>;<br/></div></div>
