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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">a</font>, <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><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 <font color="Olive" title="b2">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <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="b3">b</font></span>)</span> holds <br/> <a href="uproots.html#K6" title="UPROOTS:func.6">multiplicity</a> (<span class="p1"><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="b3">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>,<font color="Olive" title="b2">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/><span class="kw">let </span><font color="Maroon" title="c2">a</font>, <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" 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 <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <font color="Olive" title="b1">b</font></span>)</span> holds <br/> <a href="uproots.html#K6" title="UPROOTS:func.6">multiplicity</a> (<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Olive" title="b1">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>,<font color="Maroon" title="c3">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </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"> ( <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <font color="Maroon" title="c4">b</font></span>)</span> implies  <a href="uproots.html#K6" title="UPROOTS:func.6">multiplicity</a> (<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>,<font color="Maroon" title="c3">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a>  )</span><br/>





<span class="kw">assume </span><a NAME="E1:21"/><span class="lab"><font color="Green" title="E11">A1</font></span>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="algstr_0.html#K4" title="ALGSTR_0:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">a</font> <a href="algstr_0.html#K12" title="ALGSTR_0:func.12">/</a> <font color="Maroon" title="c4">b</font></span>)</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">  <a href="uproots.html#K6" title="UPROOTS:func.6">multiplicity</a> (<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>,<font color="Maroon" title="c3">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> </span><br/>

<span class="kw">set </span><font color="Maroon" title="c5">f</font> = <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>;<br/>
<a NAME="E2:21"/>
 <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="c2">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="c2">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/>
<a NAME="E3:21"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">x</font> <a href="tarski.html#R2" title="TARSKI:pred.2">in</a>  <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="c2">a</font>,<font color="Maroon" title="c4">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="txt" href="#E1:21"><span class="lab"><font color="Green" title="E11">A1</font></span></a>, <a class="ref" href="tarski.html#D1" onmouseover="rs('tarski/D1')" onmouseout="rh()">TARSKI:def 1</a></span>;<br/>
<a NAME="E4:21"/><span class="kw">then </span>
 not <font color="Maroon" title="c3">x</font> <a href="polynom5.html#R1" title="POLYNOM5:pred.1">is_a_root_of</a> <span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>
 
<span class="kw">by</span> <span class="lab"><a class="ref" href="polynom5.html#D10" onmouseover="rs('polynom5/D10')" onmouseout="rh()">POLYNOM5:def 10</a></span>;<br/>
<span class="kw">hence </span><a NAME="E5:21"/>
 <a href="uproots.html#K6" title="UPROOTS:func.6">multiplicity</a> (<span class="p1"><a href="polynom5.html#K5" title="POLYNOM5:func.5">&lt;%</a><span class="default"><font color="Maroon" title="c2">a</font>,<font color="Maroon" title="c4">b</font></span><a href="polynom5.html#K5" title="POLYNOM5:func.5">%&gt;</a></span>,<font color="Maroon" title="c3">x</font>) <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">0</a> 
 <span class="kw">by</span> <span class="lab"><a class="ref" href="nat_1.html#T14" onmouseover="rs('nat_1/T14')" onmouseout="rh()">NAT_1:14</a>, <a class="ref" href="uproots.html#T52" onmouseover="rs('uproots/T52')" onmouseout="rh()">UPROOTS:52</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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