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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> <a href="vectsp_1.html#V9" title="VECTSP_1:attr.9">left_unital</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">n</font> be  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a>  <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ;<br/>



<b>set </b><font color="Maroon" title="c3">p</font> = <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>;<br/>
<a NAME="E1:48"/>
( <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> is   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">L</font> &amp; <font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
;<br/>
<a NAME="E2:48"/><b>then </b><i><font color="Green" title="E40">A1</font></i>: 
<font color="Maroon" title="c2">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T17" title="NORMSP_1:th.17">NORMSP_1:17</a></i>;<br/>
<b>set </b><font color="Maroon" title="c4">q</font> = <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">L</font></span>)</span>;<br/>
<a NAME="E3:48"/><i><font color="Green" title="E41">A2</font></i>: 
 <a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">n</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> ,<span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="funct_7.html#T35" title="FUNCT_7:th.35">FUNCT_7:35</a></i>;<br/>
<a NAME="E4:48"/>
( <span class="p1">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c2">n</font>,<span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">L</font></span>)</span> is   <a href="struct_0.html#NM9" title="STRUCT_0:NM.9">sequence</a> of <font color="Maroon" title="c1">L</font> &amp;  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 
;<br/>
<a NAME="E5:48"/><b>then </b>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relat_1.html#K1" title="RELAT_1:func.1">dom</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom3.html#K9" title="POLYNOM3:func.9">0_.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="polynom1.html#K1" title="POLYNOM1:func.1">+*</a> <font color="Maroon" title="c2">n</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">L</font></span>)</span></span>)</span>
 
<b>by </b><i><a class="ref" href="normsp_1.html#T17" title="NORMSP_1:th.17">NORMSP_1:17</a></i>;<br/>
<b>hence </b><a NAME="E6:48"/>
( <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">L</font></span>)</span> &amp; <span class="p1">(<span class="default"><a href="uniroots.html#K4" title="UNIROOTS:func.4">unital_poly</a> <font color="Maroon" title="c1">L</font>,<font color="Maroon" title="c2">n</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c2">n</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K2" title="GROUP_1:func.2">1_</a> <font color="Maroon" title="c1">L</font> )
 <b>by </b><i><a class="txt" href="uniroots.html#E2:48"><i><font color="Green" title="E40">A1</font></i></a>, <a class="txt" href="uniroots.html#E3:48"><i><font color="Green" title="E41">A2</font></i></a>, <a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>


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