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<b>let </b><font color="Maroon" title="c1">L</font> be  non <a href="struct_0.html#V6" title="STRUCT_0:attr.6">degenerated</a> <a href="vectsp_2.html#NM1" title="VECTSP_2:NM.1">comRing</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font>, <font color="Maroon" title="c3">q</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/><b>let </b><font color="Maroon" title="c4">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a><br/>





<b>assume </b><a NAME="E1:37"/><i><font color="Green" title="E27">A1</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c3">q</font></span>)</span>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:37_1"/>
( <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c2">p</font> or <font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c3">q</font> )
 </span><b>by </b><i><a class="txt" href="uproots.html#E1:37"><i><font color="Green" title="E27">A1</font></i></a>, <a class="ref" href="xboole_0.html#D2" title="XBOOLE_0:def.2">XBOOLE_0:def 2</a></i>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_1"><b>suppose </b></a><a NAME="E1:37_1_1"/><i><font color="Green" title="E28">A2</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c2">p</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><a NAME="E3:37_1_1"/>
<font color="Maroon" title="c5">a</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c2">p</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:37_1_1"><i><font color="Green" title="E28">A2</font></i></a>, <a class="ref" href="polynom5.html#D9" title="POLYNOM5:def.9">POLYNOM5:def 9</a></i>;<br/><a NAME="E4:37_1_1"/><b>then </b>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</a></i>;<br/><a NAME="E5:37_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></i>;<br/><a NAME="E6:37_1_1"/><b>then </b>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="polynom4.html#T27" title="POLYNOM4:th.27">POLYNOM4:27</a></i>;<br/><a NAME="E7:37_1_1"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c2">p</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">q</font>
 <b>by </b><i><a class="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</a></i>;<br/><b>hence </b><a NAME="E8:37_1_1"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#D9" title="POLYNOM5:def.9">POLYNOM5:def 9</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="37_1_2"><b>suppose </b></a><a NAME="E1:37_1_2"/><i><font color="Green" title="E28">A3</font></i>: 
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c3">q</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon" title="c5">a</font> = <font color="Maroon" title="c4">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> ;<br/><a NAME="E3:37_1_2"/>
<font color="Maroon" title="c5">a</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c3">q</font>
 <b>by </b><i><a class="txt" href="uproots.html#E1:37_1_2"><i><font color="Green" title="E28">A3</font></i></a>, <a class="ref" href="polynom5.html#D9" title="POLYNOM5:def.9">POLYNOM5:def 9</a></i>;<br/><a NAME="E4:37_1_2"/><b>then </b>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</a></i>;<br/><a NAME="E5:37_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c2">p</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">q</font>,<font color="Maroon" title="c5">a</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>;<br/><a NAME="E6:37_1_2"/><b>then </b>
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>,<font color="Maroon" title="c5">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
 <b>by </b><i><a class="ref" href="polynom4.html#T27" title="POLYNOM4:th.27">POLYNOM4:27</a></i>;<br/><a NAME="E7:37_1_2"/><b>then </b>
<font color="Maroon" title="c5">a</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c2">p</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">q</font>
 <b>by </b><i><a class="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</a></i>;<br/><b>hence </b><a NAME="E8:37_1_2"/>
<font color="Maroon" title="c4">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="polynom3.html#K12" title="POLYNOM3:func.12">*'</a> <font color="Maroon" title="c3">q</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#D9" title="POLYNOM5:def.9">POLYNOM5:def 9</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

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