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<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font>) <b>-&gt; </b>   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  =  <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">p</font>,<font color="Maroon">a<sub>1</sub></font>;<br/>
<b>consider </b><font color="Maroon">b</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>, <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E2:75_1_1"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive">x</font>)
 <b>from </b><i><a class="ref" href="funct_2.html#S4">FUNCT_2:sch 4</a>();<br/></i>
<a NAME="E3:75_1_1"/><i><font color="Green">E43</font></i>: 
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E4:75_1_1"/><b>then </b><i><font color="Green">E44</font></i>: 
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font>
 
<b>by </b><i><a class="ref" href="polynom1.html#T41">POLYNOM1:41</a></i>;<br/>
<div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="75_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:75_1_1_1_1"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 ;<br/><a NAME="E2:75_1_1_1_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/><b>reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> <b>by </b><i>, </i>;<br/><a NAME="E4:75_1_1_1_1"/><i><font color="Green">E50</font></i>: 
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a> <font color="Maroon">H<sub>1</sub></font>(<font color="Maroon">xx</font>)
 <b>by </b><i/>;<br/><a NAME="E5:75_1_1_1_1"/>
0 <a href="xxreal_0.html#R1">&lt;</a> <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">xx</font>
 <b>by </b><i/>;<br/><a NAME="E6:75_1_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">xx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="binop_2.html#K23">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E7:75_1_1_1_1"/><b>then </b>
<font color="Maroon">xx</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">p</font>
 <b>by </b><i>, </i>;<br/><b>hence </b><a NAME="E8:75_1_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6">Roots</a> <font color="Maroon">p</font>
 <b>by </b><i><a class="ref" href="polynom5.html#D9">POLYNOM5:def 9</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:75_1_1_1"/><i><font color="Green">E51</font></i>: 
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom5.html#K6">Roots</a> <font color="Maroon">p</font>
 ;<br/><b>then reconsider </b><font color="Maroon">xx</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> ;<br/><a NAME="E4:75_1_1_1"/>
<font color="Maroon">xx</font> <a href="polynom5.html#R1">is_a_root_of</a> <font color="Maroon">p</font>
 <b>by </b><i>, <a class="ref" href="polynom5.html#D9">POLYNOM5:def 9</a></i>;<br/><a NAME="E5:75_1_1_1"/><b>then </b>
 <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">p</font>,<font color="Maroon">xx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i/>;<br/><a NAME="E6:75_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">xx</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 1
 <b>by </b><i/>;<br/><a NAME="E7:75_1_1_1"/><b>then </b>
<font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Maroon">xx</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><b>hence </b><a NAME="E8:75_1_1_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font>
 <b>by </b><i><a class="ref" href="polynom1.html#D7">POLYNOM1:def 7</a></i>;<br/></div><b>end;</b></div>
<a NAME="E6:75_1_1"/><b>then </b>
 <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="polynom5.html#K6">Roots</a> <font color="Maroon">p</font>
 
<b>by </b><i><a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>
<b>then reconsider </b><font color="Maroon">b</font> = <font color="Maroon">b</font> as   <a href="polynom1.html#NM1">bag</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">L</font> <b>by </b><i>, <a class="ref" href="pboole.html#D3">PBOOLE:def 3</a>, <a class="ref" href="polynom1.html#D8">POLYNOM1:def 8</a></i>;<br/>
<b>take </b>
<font color="Maroon">b</font>
;<br/>

<b>thus </b><a NAME="E8:75_1_1"/>
(  <a href="polynom1.html#K11">support</a> <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="polynom5.html#K6">Roots</a> <font color="Maroon">p</font> &amp; ( for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">L</font> holds  <font color="Maroon">b</font> <a href="polynom1.html#K8">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="uproots.html#K7" target="_self">multiplicity</a> <font color="Maroon">p</font>,<font color="Olive">x</font> ) )
 <b>by </b><i>, , <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


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