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<div><span class="kw">theorem </span><span class="lab"><font color="Green" title="E23">Th25</font></span>: <a NAME="T25"><span class="comment"><font color="firebrick">:: ZMODUL03:25</font></span><br/></a><div class="add"> for <font color="Olive" title="b1">p</font> being   <a href="int_2.html#V1" title="INT_2:attr.1">prime</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <a href="int_3.html#K1" title="INT_3:func.1">INT.Ring</a><br/>  for <font color="Olive" title="b2">V</font> being   <a href="vectsp_7.html#V3" title="VECTSP_7:attr.3">free</a>  <a href="zmodul01.html#NM1" title="ZMODUL01:NM.1">Z_Module</a><br/>  for <font color="Olive" title="b3">I</font> being   <a href="zmodul03.html#NM1" title="ZMODUL03:NM.1">Basis</a> of <font color="Olive" title="b2">V</font><br/>  for <font color="Olive" title="b4">X</font> being   non  <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <span class="p1">(<span class="default"><a href="zmodul02.html#K7" title="ZMODUL02:func.7">Z_MQ_VectSp</a> (<font color="Olive" title="b2">V</font>,<font color="Olive" title="b1">p</font>)</span>)</span>  st <font color="Olive" title="b4">X</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"> { <span class="default"> <span class="p2">(<span class="default"><a href="zmodul02.html#K8" title="ZMODUL02:func.8">ZMtoMQV</a> (<font color="Olive" title="b2">V</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b5">u</font>)</span>)</span> where <font color="Olive" title="b5">u</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="b5">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">I</font> </span> } </span>  holds <br/> ex <font color="Olive" title="b5">F</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Olive" title="b4">X</font>, the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Olive" title="b2">V</font> st <br/>( (  for <font color="Olive" title="b6">u</font> being   <a href="vectsp_1.html#NM8" title="VECTSP_1:NM.8">Vector</a> of <font color="Olive" title="b2">V</font>  st <font color="Olive" title="b6">u</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a> <font color="Olive" title="b3">I</font> holds <br/><font color="Olive" title="b5">F</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="zmodul02.html#K8" title="ZMODUL02:func.8">ZMtoMQV</a> (<font color="Olive" title="b2">V</font>,<font color="Olive" title="b1">p</font>,<font color="Olive" title="b6">u</font>)</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b6">u</font> ) &amp; <font color="Olive" title="b5">F</font> is  <a href="funct_1.html#V2" title="FUNCT_1:attr.2">one-to-one</a>  &amp;  <a href="relset_1.html#K1" title="RELSET_1:func.1">dom</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b4">X</font> &amp;  <a href="relset_1.html#K2" title="RELSET_1:func.2">rng</a> <font color="Olive" title="b5">F</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b3">I</font> )</div></div>
