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<div><span class="kw">theorem </span><a NAME="T40"><span class="comment"><font color="firebrick">:: ZMODUL07:40</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">W1</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="b4">W2</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of <font color="Olive" title="b3">W1</font><br/>  for <font color="Olive" title="b5">U1</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">U2</font> being   <a href="zmodul01.html#NM3" title="ZMODUL01:NM.3">Submodule</a> of  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Olive" title="b2">V</font>,<font color="Olive" title="b5">U1</font>)  st <font color="Olive" title="b5">U1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">W2</font> &amp; <font color="Olive" title="b6">U2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Olive" title="b3">W1</font>,<font color="Olive" title="b4">W2</font>) holds <br/> ex <font color="Olive" title="b7">F</font> being   <a href="ranknull.html#NM1" title="RANKNULL:NM.1">linear-transformation</a> of <span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<span class="p2">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Olive" title="b2">V</font>,<font color="Olive" title="b5">U1</font>)</span>)</span>,<font color="Olive" title="b6">U2</font>)</span>)</span>,<span class="p1">(<span class="default"><a href="vectsp10.html#K6" title="VECTSP10:func.6">VectQuot</a> (<font color="Olive" title="b2">V</font>,<font color="Olive" title="b3">W1</font>)</span>)</span> st <font color="Olive" title="b7">F</font> is  <a href="funct_2.html#V3" title="FUNCT_2:attr.3">bijective</a> </div></div>
