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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E20">ZM05Th59</font></span>: <a NAME="T21"><span class="comment"><font color="firebrick">:: ZMODUL07:21</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">W</font>, <font color="Olive" title="b2">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="b3">A</font> being   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">V</font><br/>  for <font color="Olive" title="b4">B</font> being   <a href="vectsp_7.html#V1" title="VECTSP_7:attr.1">linearly-independent</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Olive" title="b2">V</font><br/>  for <font color="Olive" title="b5">T</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <font color="Olive" title="b2">V</font>,<font color="Olive" title="b1">W</font><br/>  for <font color="Olive" title="b6">l</font> being    <a href="vectsp_6.html#M2" title="VECTSP_6:mode.2">Linear_Combination</a> of <font color="Olive" title="b4">B</font> <a href="subset_1.html#K7" title="SUBSET_1:func.7">\</a> <font color="Olive" title="b3">A</font>  st  <a href="zmodul03.html#K1" title="ZMODUL03:func.1">rank</a> <font color="Olive" title="b2">V</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <font color="Olive" title="b4">B</font> &amp; <font color="Olive" title="b3">A</font> is   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <span class="p1">(<span class="default"><a href="ranknull.html#K1" title="RANKNULL:func.1">ker</a> <font color="Olive" title="b5">T</font></span>)</span> &amp; <font color="Olive" title="b3">A</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Olive" title="b4">B</font> holds <br/><font color="Olive" title="b5">T</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <font color="Olive" title="b6">l</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp_6.html#K4" title="VECTSP_6:func.4">Sum</a> <span class="p1">(<span class="default"><font color="Olive" title="b5">T</font> <a href="zmodul05.html#K5" title="ZMODUL05:func.5">@*</a> <font color="Olive" title="b6">l</font></span>)</span></div></div>
