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<span class="kw">let </span><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>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font><br/> for <font color="Olive" title="b2">p</font>, <font color="Olive" title="b3">q</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b2">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Olive" title="b1">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Olive" title="b3">q</font> holds <br/><font color="Olive" title="b1">r</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Olive" title="b2">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c2">r</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> for <font color="Olive" title="b1">p</font>, <font color="Olive" title="b2">q</font> being  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>  st <font color="Olive" title="b1">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Olive" title="b2">q</font> holds <br/><font color="Maroon" title="c2">r</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Olive" title="b1">p</font></span><br/><span class="kw">let </span><font color="Maroon" title="c3">p</font>, <font color="Maroon" title="c4">q</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:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> ( <font color="Maroon" title="c3">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c4">q</font> implies <font color="Maroon" title="c2">r</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c3">p</font> )</span><br/>





<span class="kw">assume </span><a NAME="E1:72"/>
<font color="Maroon" title="c3">p</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <span class="p1"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">r</font></span>)</span>,<span class="p2">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span> <a href="polynom3.html#K13" title="POLYNOM3:func.13">*'</a> <font color="Maroon" title="c4">q</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> <font color="Maroon" title="c2">r</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c3">p</font></span><br/>

<span class="kw">then </span> <a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c3">p</font>,<font color="Maroon" title="c2">r</font> = 
<span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <span class="p2"><a href="polynom5.html#K4" title="POLYNOM5:func.4">&lt;%</a><span class="default"><span class="p3">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">r</font></span>)</span>,<span class="p3">(<span class="default"><a href="struct_0.html#K5" title="STRUCT_0:func.5">1.</a> <font color="Maroon" title="c1">L</font></span>)</span></span><a href="polynom5.html#K4" title="POLYNOM5:func.4">%&gt;</a></span>,<font color="Maroon" title="c2">r</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c2">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom4.html#T27" title="POLYNOM4:th.27">POLYNOM4:27</a></span>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="rlvect_1.html#K3" title="RLVECT_1:func.3">+</a> <font color="Maroon" title="c2">r</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c2">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#T48" title="POLYNOM5:th.48">POLYNOM5:48</a></span>
<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#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom4.html#K2" title="POLYNOM4:func.2">eval</a> <font color="Maroon" title="c4">q</font>,<font color="Maroon" title="c2">r</font></span>)</span>
<span class="kw">by</span> <span class="lab"><a class="ref" href="rlvect_1.html#D13" title="RLVECT_1:def.13">RLVECT_1:def 13</a></span>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">L</font>
<span class="kw">by</span> <span class="lab"><a class="ref" href="vectsp_1.html#T39" title="VECTSP_1:th.39">VECTSP_1:39</a></span>
;<br/>
<span class="kw">hence </span><a NAME="E3:72"/>
<font color="Maroon" title="c2">r</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <font color="Maroon" title="c3">p</font>
 <span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D6" title="POLYNOM5:def.6">POLYNOM5:def 6</a></span>; <a class="txt" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="Red">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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