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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/>for <font color="Olive" title="b2">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b2">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Olive" title="b1">p</font> iff <font color="Olive" title="b2">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="Olive" title="b1">p</font> )</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 for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c2">p</font> iff <font color="Olive" title="b1">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> ) )</span><br/>



<b>assume </b><a NAME="E1:77"/><i><font color="Green" title="E55">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"> for <font color="Olive" title="b1">x</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font> holds <br/> ( <font color="Olive" title="b1">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c2">p</font> iff <font color="Olive" title="b1">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> )</span><br/>

<a NAME="E2:77"/><b>then </b>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR4" title="XXREAL_0:NR.4">&gt;</a>  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a> 
 
;<br/>
<a NAME="E3:77"/><b>then </b>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font> <a href="xxreal_0.html#NR2" title="XXREAL_0:NR.2">&gt;=</a> <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E4:77"/><b>then </b>
 <a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</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="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1
 
<b>by </b><i><a class="ref" href="binarith.html#T53" title="BINARITH:th.53">BINARITH:53</a></i>;<br/>
<a NAME="E5:77"/><b>then </b>
<font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="algseq_1.html#T25" title="ALGSEQ_1:th.25">ALGSEQ_1:25</a></i>;<br/>
<a NAME="E6:77"/><b>then </b><i><font color="Green" title="E56">A2</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a>  <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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#T74" title="VECTSP_1:th.74">VECTSP_1:74</a></i>;<br/>
<b>let </b><font color="Maroon" title="c3">x</font> be   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</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"> ( <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> iff <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> )</span><br/>

<b>thus </b><a NAME="E7:77"/>
( <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> implies <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> )
  <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <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> implies <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> )</span><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="77_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:77_1"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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></span><br/>

<a NAME="E2:77_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="c3">x</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" target="_self" title="POLYNOM5:def.6">Def6</a></i>;<br/>
<b>then </b> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c3">x</font> = 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E1:77"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="polynom5.html#T59" target="_self" title="POLYNOM5:th.59">Th59</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D23" title="VECTSP_1:def.23">VECTSP_1:def 23</a></i>
<br/>.= 
 <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/>
<b>hence </b><a NAME="E4:77_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="ref" href="polynom5.html#D6" target="_self" title="POLYNOM5:def.6">Def6</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="E8:77"/>
<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>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <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></span><br/>

<b>then </b> <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font> = 
 <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</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>,<font color="Maroon" title="c3">x</font>
<b>by </b><i><a class="ref" href="polynom5.html#D6" target="_self" title="POLYNOM5:def.6">Def6</a></i>
<br/>.= 
<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="c3">x</font></span>)</span> <a href="vectsp_1.html#K3" title="VECTSP_1:func.3">/</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="polynom5.html#E1:77"><i><font color="Green" title="E55">A1</font></i></a>, <a class="ref" href="polynom5.html#T59" target="_self" title="POLYNOM5:th.59">Th59</a></i>
<br/>.= 
<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="c3">x</font></span>)</span> <a href="group_1.html#K9" title="GROUP_1:func.9">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c2">p</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K1" title="ALGSEQ_1:func.1">len</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="binarith.html#K3" title="BINARITH:func.3">-'</a> 1</span>)</span></span>)</span> <a href="algstr_0.html#K11" title="ALGSTR_0:func.11">"</a> </span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#D23" title="VECTSP_1:def.23">VECTSP_1:def 23</a></i>
;<br/>
<a NAME="E10:77"/><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="c3">x</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="txt" href="polynom5.html#E6:77"><i><font color="Green" title="E56">A2</font></i></a>, <a class="ref" href="vectsp_1.html#T44" title="VECTSP_1:th.44">VECTSP_1:44</a></i>;<br/>
<b>hence </b><a NAME="E11:77"/>
<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#D6" target="_self" title="POLYNOM5:def.6">Def6</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>
