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<span class="kw">let </span><font color="Maroon" title="c1">L</font> be   <a href="vectsp_1.html#NM5" title="VECTSP_1:NM.5">Field</a>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">p</font> being   <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font><br/>  for <font color="Olive" title="b2">a</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="b3">b</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st  not  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b2">a</font> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <font color="Olive" title="b3">b</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Olive" title="b1">p</font> holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Olive" title="b2">a</font>,<font color="Olive" title="b3">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Olive" title="b1">p</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c2">p</font> be   <a href="uproots.html#V1" title="UPROOTS:attr.1">non-zero</a>  <a href="polynom3.html#NM1" title="POLYNOM3:NM.1">Polynomial</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">a</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">b</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st  not  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">a</font> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c2">p</font> holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Olive" title="b1">a</font>,<font color="Olive" title="b2">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c3">a</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" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide">  for <font color="Olive" title="b1">b</font> being   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>  st  not  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <font color="Olive" title="b1">b</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c2">p</font> holds <br/> <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Olive" title="b1">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span></span><br/><span class="kw">let </span><font color="Maroon" title="c4">b</font> be   non  <a href="struct_0.html#V9" title="STRUCT_0:attr.9">zero</a>  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">L</font>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> (  not  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c2">p</font> implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> )</span><br/>







<a NAME="E1:22"/><span class="lab"><font color="Green" title="E12">A1</font></span>: 
 <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p1">(<span class="default"><span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c2">p</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p2"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span></span>)</span> <a href="subset_1.html#K4" title="SUBSET_1:func.4">\/</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c2">p</font></span>)</span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="uproots.html#T23" onmouseover="rs('uproots/T23')" onmouseout="rh()">UPROOTS:23</a></span>;<br/>
<a NAME="E2:22"/>
 <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <font color="Maroon" title="c4">b</font></span>)</span></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polyvie1.html#T10" target="_self" onmouseover="rs('polyvie1/T10')" onmouseout="rh()">Th10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E3:22"/>
(  not  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c3">a</font> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <font color="Maroon" title="c4">b</font></span>)</span> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c2">p</font> implies  <a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p1">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <span class="p2">(<span class="default"><span class="p3"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c3">a</font>,<font color="Maroon" title="c4">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span> <a href="polynom3.html#K11" title="POLYNOM3:func.11">*'</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> 1 <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> <span class="p1">(<span class="default"><a href="card_1.html#K4" title="CARD_1:func.4">card</a> <span class="p2">(<span class="default"><a href="polynom5.html#K7" title="POLYNOM5:func.7">Roots</a> <font color="Maroon" title="c2">p</font></span>)</span></span>)</span> )
 <span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:22"><span class="lab"><font color="Green" title="E12">A1</font></span></a>, <a class="ref" href="card_2.html#T41" onmouseover="rs('card_2/T41')" onmouseout="rh()">CARD_2:41</a></span>; <a class="txt" onmouseover="tooltip.show('hs',this)" onmouseout="tooltip.hide()" onclick="hs(this)" href="javascript:()"><span class="comment"><font color="firebrick">::  thesis: </font></span></a><span class="hide"> verum</span><br/>


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