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<b>let </b><font color="Maroon" title="c1">L</font> be   <a href="vectsp_1.html#NM3" title="VECTSP_1:NM.3">Field</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">p</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>  st  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Olive" title="b1">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  holds <br/>( <font color="Olive" title="b1">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> iff  <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Olive" title="b1">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> )</span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  implies ( <font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> iff  <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> ) )</span><br/>



<b>assume </b><a NAME="E1:78"/><i><font color="Green" title="E56">A1</font></i>: 
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> iff  <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> )</span><br/>

<b>thus </b><a NAME="E2:78"/>
( <font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> implies  <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> implies <font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="78_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:78_1"/>
<font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a></span><br/>

<b>then consider </b><font color="Maroon" title="c3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E3:78_1"/><i><font color="Green" title="E57">A2</font></i>: 
<font color="Maroon" title="c3">x</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="ref" href="polynom5.html#D7" target="_self" title="POLYNOM5:def.7">Def7</a></i>;<br/>
<a NAME="E4:78_1"/>
<font color="Maroon" title="c3">x</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#E1:78"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="polynom5.html#E3:78_1"><i><font color="Green" title="E57">A2</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="E5:78_1"/>
 <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a>
 <b>by </b><i><a class="ref" href="polynom5.html#D7" target="_self" title="POLYNOM5:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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

<b>assume </b><a NAME="E3:78"/>
 <a href="polynom5.html#K7" title="POLYNOM5:func.7">NormPolynomial</a> <font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a></span><br/>

<b>then consider </b><font color="Maroon" title="c3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><b> such that </b><br/><a NAME="E5:78"/><i><font color="Green" title="E57">A3</font></i>: 
<font color="Maroon" title="c3">x</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="ref" href="polynom5.html#D7" target="_self" title="POLYNOM5:def.7">Def7</a></i>;<br/>
<a NAME="E6:78"/>
<font color="Maroon" title="c3">x</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#E1:78"><i><font color="Green" title="E56">A1</font></i></a>, <a class="txt" href="polynom5.html#E5:78"><i><font color="Green" title="E57">A3</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="E7:78"/>
<font color="Maroon" title="c2">p</font> is <a href="polynom5.html#V1" title="POLYNOM5:attr.1">with_roots</a>
 <b>by </b><i><a class="ref" href="polynom5.html#D7" target="_self" title="POLYNOM5:def.7">Def7</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


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