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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="vectsp_1.html#NM2" title="VECTSP_1:NM.2">Field</a>;<br/><b>let </b><font color="Maroon" title="c2">p</font> be   <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>;<br/>



<b>assume </b><a NAME="E1:79"/><i><font color="Green" title="E56">A1</font></i>: 
 <a href="algseq_1.html#K3" title="ALGSEQ_1:func.3">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a> 0
 ;<br/>

<b>thus </b><a NAME="E2:79"/>
 <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c2">p</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c2">p</font></span>)</span>
  <i><font color="Red">:: according to </font></i><a class="ref" href="xboole_0.html#D10" title="XBOOLE_0:def.10">XBOOLE_0:def 10</a><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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="E1:79_1"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c3">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/>

<b>then reconsider </b><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c3">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="E3:79_1"/>
<font color="Maroon" title="c4">x1</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="polynom5.html#E1:79_1"><i><font color="Green" title="E57">A2</font></i></a>, <a class="ref" href="polynom5.html#D9" target="_self" title="POLYNOM5:def.9">Def9</a></i>;<br/>
<a NAME="E4:79_1"/><b>then </b>
<font color="Maroon" title="c4">x1</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="polynom5.html#E1:79"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="polynom5.html#T60" target="_self" title="POLYNOM5:th.60">Th60</a></i>;<br/>
<b>hence </b><a NAME="E5:79_1"/>
<font color="Maroon" title="c3">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"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c2">p</font></span>)</span>
 <b>by </b><i><a class="ref" href="polynom5.html#D9" target="_self" title="POLYNOM5:def.9">Def9</a></i>;<br/>


</div><b>end;</b></div>

<b>thus </b><a NAME="E3:79"/>
 <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="polynom5.html#K6" title="POLYNOM5:func.6">Roots</a> <font color="Maroon" title="c2">p</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="79_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c3">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="E1:79_2"/><i><font color="Green" title="E57">A3</font></i>: 
<font color="Maroon" title="c3">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"><a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c2">p</font></span>)</span>
 ;<br/>

<b>then reconsider </b><font color="Maroon" title="c4">x1</font> = <font color="Maroon" title="c3">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="E3:79_2"/>
<font color="Maroon" title="c4">x1</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c2">p</font>
 
<b>by </b><i><a class="txt" href="polynom5.html#E1:79_2"><i><font color="Green" title="E57">A3</font></i></a>, <a class="ref" href="polynom5.html#D9" target="_self" title="POLYNOM5:def.9">Def9</a></i>;<br/>
<a NAME="E4:79_2"/><b>then </b>
<font color="Maroon" title="c4">x1</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="polynom5.html#E1:79"><i><font color="Green" title="E56">A1</font></i></a>, <a class="ref" href="polynom5.html#T60" target="_self" title="POLYNOM5:th.60">Th60</a></i>;<br/>
<b>hence </b><a NAME="E5:79_2"/>
<font color="Maroon" title="c3">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>
 <b>by </b><i><a class="ref" href="polynom5.html#D9" target="_self" title="POLYNOM5:def.9">Def9</a></i>;<br/>


</div><b>end;</b></div>


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