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<div class="add">

<b>set </b><font color="Maroon">PR</font> =  <a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font>;<br/>
<b>set </b><font color="Maroon">cPR</font> = the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>;<br/>
<b>set </b><font color="Maroon">cR</font> = the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_1_1"/>
not  <a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font> is <a href="ideal_1.html#V7">Noetherian</a>
 ;<br/><b>then consider </b><font color="Maroon">I</font> being   <a href="ideal_1.html#NM1">Ideal</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E3:41_1_1_1"/><i><font color="Green">E44</font></i>: 
not <font color="Maroon">I</font> is <a href="ideal_1.html#V6">finitely_generated</a>
 <b>by </b><i><a class="ref" href="ideal_1.html#D27">IDEAL_1:def 27</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>( <font color="Maroon">a<sub>2</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> implies  ex <font color="Olive">A</font>, <font color="Olive">B</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> &amp;  ex <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Olive">B</font> &amp; <font color="Maroon">a<sub>3</sub></font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">r</font></span><a href="domain_1.html#K6">}</a></span> ) ) );<br/><div><i><font color="Green">E45</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">x</font> be   <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>;<br/><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_1_1_1"/>
(  not <font color="Maroon">x</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> or <font color="Maroon">x</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1_1_1_1"/>
<font color="Maroon">x</font> is  not  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_1_1_1_1_1"/>
 ex <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_1_1_1_2"/><i><font color="Green">E46</font></i>: 
<font color="Maroon">x</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font>
 ;<br/><div class="add"><b>then reconsider </b><font color="Maroon">x'</font> = <font color="Maroon">x</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> ;<br/><b>set </b><font color="Maroon">B</font> = <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_1_1_2_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_1_1_1_1_2_1"/>
<font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> <a href="hidden.html#R1">=</a>  <a href="xboole_0.html#K1">{}</a> 
 ;<br/><a NAME="E2:41_1_1_1_1_1_2_1"/><b>then </b><i><font color="Green">E47</font></i>: 
<font color="Maroon">I</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">x'</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="xboole_1.html#T37">XBOOLE_1:37</a></i>;<br/><a NAME="E3:41_1_1_1_1_1_2_1"/>
<font color="Maroon">x'</font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, <a class="ref" href="ideal_1.html#T57">IDEAL_1:57</a></i>;<br/><a NAME="E4:41_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T44">IDEAL_1:44</a></i>;<br/><a NAME="E5:41_1_1_1_1_1_2_1"/><b>then </b>
<font color="Maroon">x'</font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D10">XBOOLE_0:def 10</a></i>;<br/><b>hence </b><a NAME="E6:41_1_1_1_1_1_2_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, <a class="ref" href="ideal_1.html#D26">IDEAL_1:def 26</a></i>;<br/></div><b>end;</b></div><b>then reconsider </b><font color="Maroon">B</font> = <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">x'</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> ;<br/><a NAME="E5:41_1_1_1_1_1_2"/><i><font color="Green">E48</font></i>: 
 <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Maroon">B</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E7:41_1_1_1_1_1_2"/><i><font color="Green">E49</font></i>: 
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Maroon">B</font>
 <b>by </b><i><a class="ref" href="xboole_0.html#D1">XBOOLE_0:def 1</a></i>;<br/><b>reconsider </b><font color="Maroon">r</font> = <font color="Maroon">r</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i>, </i>;<br/><a NAME="E9:41_1_1_1_1_1_2"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Maroon">x'</font> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span>]
 <b>by </b><i/>;<br/><b>hence </b><a NAME="E10:41_1_1_1_1_1_2"/>
 ex <font color="Olive">y</font> being  <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> st <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">n</font>,<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>consider </b><font color="Maroon">F</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> , <a href="zfmisc_1.html#K1">bool</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E6:41_1_1_1"/><i><font color="Green">E50</font></i>: 
( <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <span class="p3">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; ( for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">n</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font>,<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>] ) )
 <b>from </b><i><a class="ref" href="recdef_1.html#S2">RECDEF_1:sch 2</a>();<br/></i><b>defpred </b><font color="Maroon">S<sub>2</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">a<sub>1</sub></font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font>;<br/><a NAME="E7:41_1_1_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a><br/>

<b>assume </b><a NAME="E1:41_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0
 ;<br/>

<a NAME="E2:41_1_1_1_2"/><b>then </b>
<font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E3:41_1_1_1_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T2">IDEAL_1:2</a></i>;<br/>


</div><b>end;</b></div><a NAME="E8:41_1_1_1"/><b>then </b><i><font color="Green">E51</font></i>: 
<font color="Maroon">S<sub>2</sub></font>[0]
 <b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a></i>;<br/><div><i><font color="Green">E53</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">n</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:41_1_1_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font>]
 ;<br/><b>then reconsider </b><font color="Maroon">Fn</font> = <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> as  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font> ;<br/><b>consider </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">B</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E4:41_1_1_1_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> )
 <b>and </b><br/><a NAME="E5:41_1_1_1_3"/><i><font color="Green">E56</font></i>: 
 ex <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Maroon">B</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">r</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a></i>;<br/><b>consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E7:41_1_1_1_3"/><i><font color="Green">E57</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Maroon">B</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Fn</font> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/><a NAME="E8:41_1_1_1_3"/>
<font color="Maroon">r</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E9:41_1_1_1_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E10:41_1_1_1_3"/>
<font color="Maroon">S<sub>2</sub></font>[<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1]
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T8">XBOOLE_1:8</a></i>;<br/></div><b>end;</b></div><a NAME="E10:41_1_1_1"/><i><font color="Green">E59</font></i>: 
for <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>2</sub></font>[<font color="Olive">n</font>]
 <b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i><a NAME="E11:41_1_1_1"/><i><font color="Green">E75</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> holds <br/><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_4"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:41_1_1_1_4"/>
<font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font>
 ;<br/>

<b>then consider </b><font color="Maroon">m</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:41_1_1_1_4"/><i><font color="Green">E76</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">m</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E4:41_1_1_1_4"/><i><font color="Green">E77</font></i>: 
<font color="Maroon">m</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span>;<br/>
<a NAME="E5:41_1_1_1_4"/><i><font color="Green">E78</font></i>: 
<font color="Maroon">S<sub>3</sub></font>[0]
 
;<br/>
<a NAME="E6:41_1_1_1_4"/><i><font color="Green">E119</font></i>: 
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">m</font>] holds <br/><font color="Maroon">S<sub>3</sub></font>[<font color="Olive">m</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_4_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">m</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:41_1_1_1_4_1"/><i><font color="Green">E121</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span>
 ;<br/>

<a NAME="E2:41_1_1_1_4_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font>
 
<b>by </b><i/>;<br/>
<b>then consider </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">B</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E4:41_1_1_1_4_1"/>
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> )
 <b>and </b><br/><a NAME="E5:41_1_1_1_4_1"/><i><font color="Green">E138</font></i>: 
 ex <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Maroon">B</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">r</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a></i>;<br/>
<b>consider </b><font color="Maroon">r</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E7:41_1_1_1_4_1"/><i><font color="Green">E151</font></i>: 
( <font color="Maroon">r</font> <a href="hidden.html#R2">in</a>  <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Maroon">B</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Maroon">r</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i/>;<br/>
<a NAME="E8:41_1_1_1_4_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">m</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 
<b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/>
<b>hence </b><a NAME="E9:41_1_1_1_4_1"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T1">XBOOLE_1:1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E7:41_1_1_1_4"/>
for <font color="Olive">m</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">m</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="hilbasis.html#E4:41_1_1_1"><i><font color="Green">E45</font></i></a>, );<br/></i>
<b>hence </b><a NAME="E8:41_1_1_1_4"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:41_1_1_1"><i><font color="Green">E44</font></i></a>, </i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>3</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">A</font>, <font color="Olive">B</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> st <br/>( <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> &amp; <font color="Olive">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Olive">A</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> &amp; <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Olive">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font></span>)</span> <a href="xboole_0.html#K2">\/</a> <span class="p1"><a href="tarski.html#K1">{</a><span class="default"><font color="Maroon">a<sub>2</sub></font></span><a href="tarski.html#K1">}</a></span> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R2">in</a>  <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Olive">B</font> );<br/><div><i><font color="Green">E152</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><a NAME="E1:41_1_1_1_5"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> is  non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font>
 <b>by </b><i/>;<br/><b>then consider </b><font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">B</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E3:41_1_1_1_5"/><i><font color="Green">E153</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> )
 <b>and </b><br/><a NAME="E4:41_1_1_1_5"/><i><font color="Green">E154</font></i>: 
 ex <font color="Olive">r</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> st <br/>( <font color="Olive">r</font> <a href="hidden.html#R2">in</a>  <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Maroon">B</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><font color="Olive">r</font></span><a href="domain_1.html#K6">}</a></span> )
 <b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a></i>;<br/><b>thus </b><a NAME="E5:41_1_1_1_5"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> st <font color="Maroon">S<sub>3</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:41_1_1_1"><i><font color="Green">E44</font></i></a>, <a class="txt" href="hilbasis.html#E4:41_1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">f</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E14:41_1_1_1"/><i><font color="Green">E155</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>3</sub></font>[<font color="Olive">x</font>,<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><a NAME="E15:41_1_1_1"/><i><font color="Green">E156</font></i>: 
for <font color="Olive">i</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   holds <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_6"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">B</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E2:41_1_1_1_6"/><i><font color="Green">E157</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:41_1_1_1"><i><font color="Green">E44</font></i></a></i>;<br/>
<a NAME="E3:41_1_1_1_6"/>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="tarski.html#R1">c=</a> <font color="Maroon">A</font> <a href="ideal_1.html#K7">-Ideal</a> 
 
<b>by </b><i>, <a class="ref" href="ideal_1.html#D15">IDEAL_1:def 15</a></i>;<br/>
<a NAME="E4:41_1_1_1_6"/><b>then </b><i><font color="Green">E158</font></i>: 
not <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 
<b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E5:41_1_1_1_6"/><i><font color="Green">E159</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0 <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 
<b>by </b><i/>;<br/>
<a NAME="E6:41_1_1_1_6"/>
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> 0
 
<b>by </b><i><a class="ref" href="hilbasis.html#T1" target="_self">Th1</a>, <a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/>
<b>hence </b><a NAME="E7:41_1_1_1_6"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><a NAME="E16:41_1_1_1"/><i><font color="Green">E185</font></i>: 
for <font color="Olive">i</font>, <font color="Olive">j</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <br/> for <font color="Olive">fi</font>, <font color="Olive">fj</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font>  st <font color="Olive">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Olive">j</font> &amp; <font color="Olive">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">i</font> &amp; <font color="Olive">fj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">j</font> holds <br/> <a href="algseq_1.html#K3">len</a> <font color="Olive">fi</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Olive">fj</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_7"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">j</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">fi</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font>, <font color="Maroon">fj</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font>;<br/>



<b>assume </b><a NAME="E1:41_1_1_1_7"/><i><font color="Green">E186</font></i>: 
( <font color="Maroon">i</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">j</font> &amp; <font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font> &amp; <font color="Maroon">fj</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">j</font> )
 ;<br/>

<b>then consider </b><font color="Maroon">k</font> being   <a href="nat_1.html#NM1">Nat</a><b> such that </b><br/><a NAME="E3:41_1_1_1_7"/><i><font color="Green">E187</font></i>: 
<font color="Maroon">j</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="xcmplx_0.html#K2">+</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="nat_1.html#T28">NAT_1:28</a></i>;<br/>
<a NAME="E4:41_1_1_1_7"/><i><font color="Green">E188</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R2">in</a>  <a href="numbers.html#K5">NAT</a> 
 
<b>by </b><i><a class="ref" href="ordinal1.html#D13">ORDINAL1:def 13</a></i>;<br/>
<b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">fk</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font>  st <font color="Olive">fk</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">a<sub>1</sub></font></span>)</span> holds <br/> <a href="algseq_1.html#K3">len</a> <font color="Maroon">fi</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Olive">fk</font>;<br/>
<a NAME="E5:41_1_1_1_7"/><i><font color="Green">E189</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[0]
 
<b>by </b><i/>;<br/>
<div><i><font color="Green">E190</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_7_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>assume </b><a NAME="E1:41_1_1_1_7_1"/><i><font color="Green">E191</font></i>: 
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">k</font>]
 ;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_7_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">fk1</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:41_1_1_1_7_1_1"/><i><font color="Green">E192</font></i>: 
<font color="Maroon">fk1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 ;<br/><b>reconsider </b><font color="Maroon">fk</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E3:41_1_1_1_7_1_1"/><i><font color="Green">E214</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">fi</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fk</font>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">B</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E5:41_1_1_1_7_1_1"/><i><font color="Green">E215</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="hidden.html#R2">in</a>  <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:41_1_1_1"><i><font color="Green">E44</font></i></a></i>;<br/><b>consider </b><font color="Maroon">n'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">A'</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">B'</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E7:41_1_1_1_7_1_1"/><i><font color="Green">E216</font></i>: 
( <font color="Maroon">A'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n'</font> &amp; <font color="Maroon">B'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A'</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> &amp; <font color="Maroon">n'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n'</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p4">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a>  <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Maroon">B'</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:41_1_1_1"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E8:41_1_1_1_7_1_1"/>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 ;<br/><a NAME="E9:41_1_1_1_7_1_1"/><b>then </b>
<font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E10:41_1_1_1_7_1_1"/><b>then </b>
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <font color="Maroon">k</font></span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
 <b>by </b><i/>;<br/><a NAME="E11:41_1_1_1_7_1_1"/><b>then </b>
<font color="Maroon">A</font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">A'</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i>, , <a class="ref" href="ideal_1.html#T57">IDEAL_1:57</a></i>;<br/><a NAME="E12:41_1_1_1_7_1_1"/><b>then </b><i><font color="Green">E217</font></i>: 
not <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">A</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E13:41_1_1_1_7_1_1"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E14:41_1_1_1_7_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">i</font> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">B</font>
 <b>by </b><i>, , <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><a NAME="E15:41_1_1_1_7_1_1"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">fk</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fk1</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E9:41_1_1_1"><i><font color="Green">E53</font></i></a>, , </i>;<br/><b>hence </b><a NAME="E16:41_1_1_1_7_1_1"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">fi</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fk1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E3:41_1_1_1_7_1"/>
<font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 ;<br/></div><b>end;</b></div>
<a NAME="E7:41_1_1_1_7"/>
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(<a class="txt" href="hilbasis.html#E6:41_1_1_1"><i><font color="Green">E50</font></i></a>, <a class="txt" href="hilbasis.html#E8:41_1_1_1"><i><font color="Green">E51</font></i></a>);<br/></i>
<b>hence </b><a NAME="E8:41_1_1_1_7"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">fi</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fj</font>
 <b>by </b><i>, , </i>;<br/>


</div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>4</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">n</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  ex <font color="Olive">A</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">n</font> &amp; <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">A</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Olive">A</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> );<br/><div><i><font color="Green">E218</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_8"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">x</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>reconsider </b><font color="Maroon">fx</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E2:41_1_1_1_8"/>
<font color="Maroon">fx</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fx</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> is    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/><b>hence </b><a NAME="E3:41_1_1_1_8"/>
 ex <font color="Olive">y</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <font color="Maroon">S<sub>4</sub></font>[<font color="Maroon">x</font>,<font color="Olive">y</font>]
 ;<br/></div><b>end;</b></div><b>consider </b><font color="Maroon">a</font> being   <a href="funct_2.html#NM1">Function</a> of  <a href="numbers.html#K5">NAT</a> ,the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E19:41_1_1_1"/><i><font color="Green">E219</font></i>: 
for <font color="Olive">x</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>4</sub></font>[<font color="Olive">x</font>,<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S3">FUNCT_2:sch 3</a>();<br/></i><b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="normsp_1.html#NM2">sequence</a> of <font color="Maroon">R</font> ;<br/><a NAME="E21:41_1_1_1"/>
for <font color="Olive">B</font> being non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>  ex <font color="Olive">C</font> being non <a href="xboole_0.html#V1">empty</a> <a href="finset_1.html#V1">finite</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">C</font> <a href="tarski.html#R1">c=</a> <font color="Olive">B</font> &amp; <font color="Olive">C</font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="hidden.html#R1">=</a> <font color="Olive">B</font> <a href="ideal_1.html#K7">-Ideal</a>  )
 <b>by </b><i><a class="ref" href="ideal_1.html#T99">IDEAL_1:99</a></i>;<br/><b>then consider </b><font color="Maroon">m</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E23:41_1_1_1"/><i><font color="Green">E220</font></i>: 
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p3">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p4">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span></span>)</span> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="ideal_1.html#T100">IDEAL_1:100</a></i>;<br/><b>consider </b><font color="Maroon">lc</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E25:41_1_1_1"/><i><font color="Green">E221</font></i>: 
<font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T60">IDEAL_1:60</a></i>;<br/><b>reconsider </b><font color="Maroon">fm1</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><div><i><font color="Green">E222</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_9"><b>now </b></a><div class="add"><b>let </b><font color="Maroon">i</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/><b>let </b><font color="Maroon">fi</font> be   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font>;<br/><b>assume </b><a NAME="E1:41_1_1_1_9"/>
<font color="Maroon">fi</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">i</font>
 ;<br/><a NAME="E2:41_1_1_1_9"/><b>then </b>
<font color="Maroon">fi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E4:41_1_1_1"><i><font color="Green">E45</font></i></a></i>;<br/><a NAME="E3:41_1_1_1_9"/><b>then </b>
<font color="Maroon">fi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a>  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E4:41_1_1_1_9"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">fi</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 <b>by </b><i><a class="ref" href="polynom4.html#T8">POLYNOM4:8</a></i>;<br/><a NAME="E5:41_1_1_1_9"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">fi</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><a NAME="E6:41_1_1_1_9"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">fi</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>hence </b><a NAME="E7:41_1_1_1_9"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fi</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="ref" href="xreal_1.html#T21">XREAL_1:21</a></i>;<br/></div><b>end;</b></div><b>defpred </b><font color="Maroon">S<sub>5</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b>(  ex <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="group_1.html#K10">*</a> <font color="Olive">v</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">w</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> ) implies  ex <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> ex <font color="Olive">p</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">a<sub>2</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> &amp; <font color="Olive">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K10">*</a> <font color="Olive">y</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">z</font> &amp; <font color="Maroon">a<sub>1</sub></font> <a href="hidden.html#R1">=</a> <font color="Olive">p</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Olive">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> ) );<br/><a NAME="E28:41_1_1_1"/><i><font color="Green">E223</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> holds <br/> ex <font color="Olive">d</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">d</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Olive">e</font>,<font color="Olive">d</font>] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_10"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">e</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:41_1_1_1_10"/>
<font color="Maroon">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_10_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_1_10_1"/>
(  ex <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="group_1.html#K10">*</a> <font color="Olive">v</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">w</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> ) or for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>  holds <br/>(  not <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="group_1.html#K10">*</a> <font color="Olive">v</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">w</font> or  not <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> ) )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_10_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1_10_1_1"/>
 ex <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <br/>( <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="group_1.html#K10">*</a> <font color="Olive">v</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">w</font> &amp; <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> )
 ;<br/><div class="add"><b>then consider </b><font color="Maroon">u</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Maroon">b'</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>, <font color="Maroon">w</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:41_1_1_1_10_1_1"/><i><font color="Green">E224</font></i>: 
<font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b'</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">w</font>
 <b>and </b><br/><a NAME="E4:41_1_1_1_10_1_1"/><i><font color="Green">E225</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 ;<br/><b>set </b><font color="Maroon">a'</font> = <font color="Maroon">u</font>;<br/><b>set </b><font color="Maroon">c'</font> = <font color="Maroon">w</font>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E6:41_1_1_1_10_1_1"/><i><font color="Green">E227</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> &amp; <font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">n</font> )
 <b>by </b><i>, <a class="ref" href="funct_1.html#D5">FUNCT_1:def 5</a></i>;<br/><i><font color="Green">E230</font></i>:  <a href="relset_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><a href="relset_1.html#K4">dom</a> <font color="Maroon">a</font></span>)</span> <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_1.html#T68">FUNCT_1:68</a></i>
<br/>.= 
<a href="numbers.html#K5">NAT</a>  <a href="subset_1.html#K9">/\</a> <span class="p1">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>
<br/>.= 
 <a href="gr_cy_1.html#K1">Segm</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="xboole_1.html#T28">XBOOLE_1:28</a></i>
;<br/><b>reconsider </b><font color="Maroon">n</font> = <font color="Maroon">n</font> as    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  <b>by </b><i/>;<br/><a NAME="E9:41_1_1_1_10_1_1"/><i><font color="Green">E232</font></i>: 
<font color="Maroon">n</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, , <a class="ref" href="gr_cy_1.html#T10">GR_CY_1:10</a></i>;<br/><b>set </b><font color="Maroon">y</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>;<br/><b>reconsider </b><font color="Maroon">y'</font> = <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>set </b><font color="Maroon">x'</font> =  <a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span>;<br/><b>set </b><font color="Maroon">z'</font> =  <a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">w</font>,0;<br/><b>reconsider </b><font color="Maroon">x</font> =  <a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span>, <font color="Maroon">z</font> =  <a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">w</font>,0 as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>set </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">z</font>;<br/><a NAME="E12:41_1_1_1_10_1_1"/><i><font color="Green">E233</font></i>: 
<font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">y'</font>
 <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E13:41_1_1_1_10_1_1"/><b>then </b><i><font color="Green">E234</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">z</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">w</font>,0</span>)</span>
 <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E14:41_1_1_1_10_1_1"/><i><font color="Green">E235</font></i>: 
<font color="Maroon">b'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font>
 <b>by </b><i>, <a class="ref" href="funct_1.html#T70">FUNCT_1:70</a></i>;<br/><b>consider </b><font color="Maroon">n'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">A</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E16:41_1_1_1_10_1_1"/><i><font color="Green">E236</font></i>: 
( <font color="Maroon">n'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n'</font> &amp; <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i/>;<br/><b>reconsider </b><font color="Maroon">p</font> = <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">z</font> as   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E18:41_1_1_1_10_1_1"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font>
 <b>by </b><i>, </i>;<br/><a NAME="E19:41_1_1_1_10_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T11">XREAL_1:11</a></i>;<br/><a NAME="E20:41_1_1_1_10_1_1"/><b>then </b><i><font color="Green">E237</font></i>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span>
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E21:41_1_1_1_10_1_1"/><i><font color="Green">E238</font></i>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="hilbasis.html#E8:41_1_1_1"><i><font color="Green">E51</font></i></a></i>;<br/><a NAME="E22:41_1_1_1_10_1_1"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="hilbasis.html#E8:41_1_1_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>then </b><span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="hilbasis.html#T3" target="_self">Th3</a>, , <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
;<br/><a NAME="E24:41_1_1_1_10_1_1"/><b>then </b><i><font color="Green">E239</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b'</font>
 <b>by </b><i><a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, , </i>;<br/><a NAME="E25:41_1_1_1_10_1_1"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 0
 ;<br/><a NAME="E26:41_1_1_1_10_1_1"/><b>then </b><i><font color="Green">E240</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">b'</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:41_1_1_1"><i><font color="Green">E59</font></i></a>, , </i>;<br/><a NAME="E27:41_1_1_1_10_1_1"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
 <b>by </b><i><a class="ref" href="hilbasis.html#T3" target="_self">Th3</a>, </i>;<br/><a NAME="E28:41_1_1_1_10_1_1"/><b>then </b><i><font color="Green">E241</font></i>: 
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span></span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E29:41_1_1_1_10_1_1"/><i><font color="Green">E242</font></i>: 
0 <a href="nat_1.html#K1">+</a> 0 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span></span>)</span>
 <b>by </b><i>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E30:41_1_1_1_10_1_1"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i/>;<br/><a NAME="E31:41_1_1_1_10_1_1"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i>, <a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E32:41_1_1_1_10_1_1"/><b>then </b><i><font color="Green">E243</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p4">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E33:41_1_1_1_10_1_1"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">w</font>,0</span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i/>;<br/><a NAME="E34:41_1_1_1_10_1_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">w</font>,0</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="hilbasis.html#T4" target="_self">Th4</a>, <a class="ref" href="xreal_1.html#T9">XREAL_1:9</a></i>;<br/><a NAME="E35:41_1_1_1_10_1_1"/><b>then </b><i><font color="Green">E244</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">w</font>,0</span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i><a class="ref" href="jordan3.html#T5">JORDAN3:5</a></i>;<br/><a NAME="E36:41_1_1_1_10_1_1"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">w</font>,0</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p5">(<span class="default"><span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p0">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">y'</font></span>)</span></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <span class="p3">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">w</font>,0</span>)</span></span>)</span></span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i/>;<br/><a NAME="E37:41_1_1_1_10_1_1"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">w</font>,0</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="binarith.html#K3">-'</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/><a NAME="E38:41_1_1_1_10_1_1"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">u</font>,<span class="p4">(<span class="default"><span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> <span class="p5">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">y'</font></span>)</span></span>)</span></span>)</span> <a href="polynom3.html#K15">*'</a> <font color="Maroon">y'</font></span>)</span> <a href="polynom3.html#K15">*'</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K5" target="_self">monomial</a> <font color="Maroon">w</font>,0</span>)</span></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="xcmplx_0.html#K6">-</a> 1
 <b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>;<br/><a NAME="E39:41_1_1_1_10_1_1"/><b>then </b><i><font color="Green">E245</font></i>: 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a> <span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="nat_1.html#K1">+</a> 0
 <b>by </b><i>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E40:41_1_1_1_10_1_1"/>
<font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1 <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E41:41_1_1_1_10_1_1"/><b>then </b><i><font color="Green">E246</font></i>: 
<font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i/>;<br/><b>consider </b><font color="Maroon">n'</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">B</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E43:41_1_1_1_10_1_1"/><i><font color="Green">E247</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n'</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> &amp; <font color="Maroon">n'</font> <a href="hidden.html#R1">=</a> <font color="Maroon">n</font> &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n'</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n'</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a>  <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:41_1_1_1"><i><font color="Green">E44</font></i></a></i>;<br/><a NAME="E44:41_1_1_1_10_1_1"/>
<span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span></span><a href="domain_1.html#K6">}</a></span> <a href="tarski.html#R1">c=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_1.html#T7">XBOOLE_1:7</a></i>;<br/><a NAME="E45:41_1_1_1_10_1_1"/><b>then </b>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="zfmisc_1.html#T37">ZFMISC_1:37</a></i>;<br/><b>hence </b><a NAME="E46:41_1_1_1_10_1_1"/>
 ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">e</font>,<font color="Olive">u</font>] )
 <b>by </b><i><a class="txt" href="hilbasis.html#E9:41_1_1_1"><i><font color="Green">E53</font></i></a>, , , </i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_10_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_1_10_1_2"/>
for <font color="Olive">u</font>, <font color="Olive">v</font>, <font color="Olive">w</font> being   <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>  holds <br/>(  not <font color="Maroon">e</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">u</font> <a href="group_1.html#K10">*</a> <font color="Olive">v</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">w</font> or  not <font color="Olive">v</font> <a href="hidden.html#R2">in</a>  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span> )
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_1_1_10_1_2"/>
 ex <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>  st <br/>( <font color="Olive">u</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> &amp; <font color="Maroon">S<sub>5</sub></font>[<font color="Maroon">e</font>,<font color="Olive">u</font>] )
 ;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div><b>consider </b><font color="Maroon">LCT</font> being   <a href="funct_2.html#NM1">Function</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>,the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E30:41_1_1_1"/><i><font color="Green">E248</font></i>: 
for <font color="Olive">e</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">e</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> holds <br/><font color="Maroon">S<sub>5</sub></font>[<font color="Olive">e</font>,<font color="Maroon">LCT</font> <a href="funct_1.html#K1">.</a> <font color="Olive">e</font>]
 <b>from </b><i><a class="ref" href="funct_2.html#S1">FUNCT_2:sch 1</a>();<br/></i><b>reconsider </b><font color="Maroon">FM1</font> = <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> as  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i/>;<br/><b>set </b><font color="Maroon">raSm1</font> =  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/><a NAME="E32:41_1_1_1"/><i><font color="Green">E249</font></i>: 
for <font color="Olive">lc</font> being   <a href="ideal_1.html#M1">LinearCombination</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>  ex <font color="Olive">LC</font> being   <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font> st <br/>( <font color="Olive">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Olive">lc</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Olive">lc</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">LC</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_11"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">lc</font> be    <a href="ideal_1.html#M1">LinearCombination</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/>

<a NAME="E1:41_1_1_1_11"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">LCT</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/>
<a NAME="E2:41_1_1_1_11"/><b>then </b>
 <a href="relset_1.html#K5">rng</a> <font color="Maroon">lc</font> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">LCT</font>
 
;<br/>
<a NAME="E3:41_1_1_1_11"/><b>then </b><i><font color="Green">E250</font></i>: 
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font></span>)</span>
 
<b>by </b><i><a class="ref" href="relat_1.html#T46">RELAT_1:46</a></i>;<br/>
<b>set </b><font color="Maroon">LC</font> = <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font>;<br/>
<a NAME="E4:41_1_1_1_11"/><i><font color="Green">E251</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font></span>)</span>
 
<b>by </b><i>, <a class="ref" href="finseq_3.html#T31">FINSEQ_3:31</a></i>;<br/>
<a NAME="E5:41_1_1_1_11"/>
<font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font> is    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_11_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">i</font> be    <a href="hidden.html#M1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D9">IDEAL_1:def 9</a><br/>

<b>assume </b><a NAME="E1:41_1_1_1_11_1"/><i><font color="Green">E252</font></i>: 
<font color="Maroon">i</font> <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <span class="p1">(<span class="default"><font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font></span>)</span>
 ;<br/>

<b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">v</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">a</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E3:41_1_1_1_11_1"/><i><font color="Green">E253</font></i>: 
<font color="Maroon">lc</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">v</font>
 <b>by </b><i>, , <a class="ref" href="ideal_1.html#D9">IDEAL_1:def 9</a></i>;<br/>
<a NAME="E4:41_1_1_1_11_1"/><i><font color="Green">E255</font></i>: 
<font color="Maroon">lc</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <font color="Maroon">lc</font> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
 
<b>by </b><i>, , <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">p</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:41_1_1_1_11_1"/><i><font color="Green">E256</font></i>: 
( <font color="Maroon">LCT</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">a</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">v</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E9:41_1_1_1"><i><font color="Green">E53</font></i></a>, </i>;<br/>
<b>reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font>, <font color="Maroon">z</font> = <font color="Maroon">z</font> as   <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> ;<br/>
<b>reconsider </b><font color="Maroon">y</font> = <font color="Maroon">y</font> as    <a href="subset_1.html#M2">Element</a> of <font color="Maroon">FM1</font> <b>by </b><i><a class="ref" href="hilbasis.html#T2" target="_self">Th2</a></i>;<br/>
<span class="p1">(<span class="default"><font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> = 
<span class="p1">(<span class="default"><font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font></span>)</span> <a href="funct_1.html#K1">.</a> <font color="Maroon">i</font>
<b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">z</font>
<b>by </b><i>, , <a class="txt" href="hilbasis.html#E10:41_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, <a class="ref" href="funct_1.html#T23">FUNCT_1:23</a></i>
;<br/>
<b>hence </b><a NAME="E10:41_1_1_1_11_1"/>
 ex <font color="Olive">x</font>, <font color="Olive">z</font> being  <a href="struct_0.html#NM1">Element</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> ex <font color="Olive">y</font> being   <a href="subset_1.html#M2">Element</a> of <font color="Maroon">FM1</font> st <span class="p1">(<span class="default"><font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font></span>)</span> <a href="finseq_4.html#K4">/.</a> <font color="Maroon">i</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Olive">x</font> <a href="group_1.html#K10">*</a> <font color="Olive">y</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Olive">z</font>
 ;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">LC</font> = <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font> as    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font> ;<br/>
<a NAME="E7:41_1_1_1_11"/>
<font color="Maroon">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LC</font>
 
;<br/>
<b>hence </b><a NAME="E8:41_1_1_1_11"/>
 ex <font color="Olive">LC</font> being   <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font> st <br/>( <font color="Olive">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Olive">LC</font> )
 <b>by </b><i/>;<br/>


</div><b>end;</b></div><a NAME="E33:41_1_1_1"/>
for <font color="Olive">lc</font> being   <a href="ideal_1.html#M1">LinearCombination</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>  ex <font color="Olive">LC</font> being   <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font> ex <font color="Olive">sLC</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Olive">lc</font> &amp; <font color="Olive">sLC</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">LC</font> &amp; <font color="Olive">sLC</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Olive">sLC</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_12"><b>proof </b></a><div class="add">

<b>defpred </b><font color="Maroon">S<sub>6</sub></font>[   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ] <b>means </b>for <font color="Olive">lc</font> being   <a href="ideal_1.html#M1">LinearCombination</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>  st  <a href="finseq_1.html#K3">len</a> <font color="Olive">lc</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">a<sub>1</sub></font> holds <br/> ex <font color="Olive">LC</font> being   <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font> ex <font color="Olive">sLC</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Olive">lc</font> &amp; <font color="Olive">sLC</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">LC</font> &amp; <font color="Olive">sLC</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">lc</font> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Olive">sLC</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font> );<br/>
<a NAME="E1:41_1_1_1_12"/><i><font color="Green">E257</font></i>: 
<font color="Maroon">S<sub>6</sub></font>[0]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_12_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">lc</font> be    <a href="ideal_1.html#M1">LinearCombination</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:41_1_1_1_12_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="xxreal_0.html#R1">&lt;=</a> 0
 ;<br/>

<a NAME="E2:41_1_1_1_12_1"/><b>then </b><i><font color="Green">E259</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> 0
 
;<br/>
<b>consider </b><font color="Maroon">LC</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font><b> such that </b><br/><a NAME="E4:41_1_1_1_12_1"/><i><font color="Green">E260</font></i>: 
( <font color="Maroon">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">LC</font> )
 <b>by </b><i/>;<br/>
<b>take </b>
<font color="Maroon">LC</font>
;<br/>

<b>take </b><font color="Maroon">p</font> =  <a href="polynom3.html#K12">0_.</a> <font color="Maroon">R</font>;<br/>

<b>thus </b><a NAME="E5:41_1_1_1_12_1"/>
<font color="Maroon">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font>
 <b>by </b><i/>;<br/>

<a NAME="E6:41_1_1_1_12_1"/><i><font color="Green">E261</font></i>: 
<font color="Maroon">lc</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 
<b>by </b><i>, <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<a NAME="E7:41_1_1_1_12_1"/>
<font color="Maroon">LC</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K6">&lt;*&gt;</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>
 
<b>by </b><i>, , <a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/>
<b>then </b> <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">LC</font> = 
 <a href="rlvect_1.html#K1">0.</a> <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>
<br/>.= 
<font color="Maroon">p</font>
<b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>
;<br/>
<b>hence </b><a NAME="E9:41_1_1_1_12_1"/>
<font color="Maroon">p</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">LC</font>
 ;<br/>

<b>thus </b><font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13">FUNCOP_1:13</a></i>
<br/>.= 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font>
<b>by </b><i>, <a class="ref" href="rlvect_1.html#T60">RLVECT_1:60</a></i>
;<br/>

<a NAME="E11:41_1_1_1_12_1"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> 0
 
<b>by </b><i><a class="ref" href="polynom4.html#T6">POLYNOM4:6</a></i>;<br/>
<b>hence </b><a NAME="E12:41_1_1_1_12_1"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:41_1_1_1_12"/><i><font color="Green">E262</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>   st <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">k</font>] holds <br/><font color="Maroon">S<sub>6</sub></font>[<font color="Olive">k</font> <a href="nat_1.html#K1">+</a> 1]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_12_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">k</font> be    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> ;<br/>

<b>assume </b><a NAME="E1:41_1_1_1_12_2"/><i><font color="Green">E263</font></i>: 
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">k</font>]
 ;<br/>

<b>thus </b><a NAME="E2:41_1_1_1_12_2"/>
<font color="Maroon">S<sub>6</sub></font>[<font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1]
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_12_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">lc</font> be    <a href="ideal_1.html#M1">LinearCombination</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/>

<b>assume </b><a NAME="E1:41_1_1_1_12_2_1"/><i><font color="Green">E264</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 ;<br/>

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_12_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_1_12_2_1_1"/>
(  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font> or  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_12_2_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1_12_2_1_1_1"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><b>hence </b><a NAME="E2:41_1_1_1_12_2_1_1_1"/>
 ex <font color="Olive">LC</font> being   <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font> ex <font color="Olive">sLC</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font> &amp; <font color="Olive">sLC</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">LC</font> &amp; <font color="Olive">sLC</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Olive">sLC</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font> )
 <b>by </b><i/>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_12_2_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_1_12_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> <font color="Maroon">k</font>
 ;<br/><div class="add"><a NAME="E2:41_1_1_1_12_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><a NAME="E3:41_1_1_1_12_2_1_1_2"/><b>then </b><i><font color="Green">E265</font></i>: 
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font> <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><b>thus </b><a NAME="E4:41_1_1_1_12_2_1_1_2"/>
 ex <font color="Olive">LC</font> being   <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font> ex <font color="Olive">sLC</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font> &amp; <font color="Olive">sLC</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">LC</font> &amp; <font color="Olive">sLC</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Olive">sLC</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font> )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_12_2_1_1_2_1"><b>proof </b></a><div class="add">

<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_12_2_1_1_2_1_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_1_12_2_1_1_2_1_1"/>
( <font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0 or <font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_12_2_1_1_2_1_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1_12_2_1_1_2_1_1_1"/><i><font color="Green">E266</font></i>: 
<font color="Maroon">k</font> <a href="hidden.html#R1">=</a> 0
 ;<br/><div class="add"><a NAME="E2:41_1_1_1_12_2_1_1_2_1_1_1"/><b>then </b>
 <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default">1</span><a href="domain_1.html#K6">}</a></span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:41_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="finseq_1.html#T4">FINSEQ_1:4</a>, <a class="ref" href="finseq_1.html#D3">FINSEQ_1:def 3</a></i>;<br/><a NAME="E3:41_1_1_1_12_2_1_1_2_1_1_1"/><b>then </b><i><font color="Green">E267</font></i>: 
1 <a href="hidden.html#R2">in</a>  <a href="finseq_1.html#K4">dom</a> <font color="Maroon">lc</font>
 <b>by </b><i><a class="ref" href="tarski.html#D1">TARSKI:def 1</a></i>;<br/><a NAME="E4:41_1_1_1_12_2_1_1_2_1_1_1"/><i><font color="Green">E268</font></i>: 
<font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="finseq_1.html#K9">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">lc</font> <a href="funct_1.html#K1">.</a> 1</span>)</span></span><a href="finseq_1.html#K9">*&gt;</a></span>
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:41_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, <a class="ref" href="finseq_1.html#T57">FINSEQ_1:57</a></i>;<br/><b>consider </b><font color="Maroon">u</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">w</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">v</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span><b> such that </b><br/><a NAME="E6:41_1_1_1_12_2_1_1_2_1_1_1"/><i><font color="Green">E269</font></i>: 
<font color="Maroon">lc</font> <a href="finseq_4.html#K4">/.</a> 1 <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">v</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">w</font>
 <b>by </b><i>, <a class="ref" href="ideal_1.html#D9">IDEAL_1:def 9</a></i>;<br/><a NAME="E7:41_1_1_1_12_2_1_1_2_1_1_1"/><i><font color="Green">E270</font></i>: 
<font color="Maroon">lc</font> <a href="funct_1.html#K1">.</a> 1 <a href="hidden.html#R1">=</a> <font color="Maroon">lc</font> <a href="finseq_4.html#K4">/.</a> 1
 <b>by </b><i>, <a class="ref" href="finseq_4.html#D4">FINSEQ_4:def 4</a></i>;<br/><b>then consider </b><font color="Maroon">x</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">y</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">z</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">p</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E9:41_1_1_1_12_2_1_1_2_1_1_1"/><i><font color="Green">E271</font></i>: 
( <font color="Maroon">LCT</font> <a href="funct_1.html#K1">.</a> <span class="p1">(<span class="default"><font color="Maroon">lc</font> <a href="funct_1.html#K1">.</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> &amp; <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">x</font> <a href="group_1.html#K10">*</a> <font color="Maroon">y</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">z</font> &amp; <span class="p1">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">v</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">w</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R2">in</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E9:41_1_1_1"><i><font color="Green">E53</font></i></a>, </i>;<br/><b>consider </b><font color="Maroon">LC</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font><b> such that </b><br/><a NAME="E11:41_1_1_1_12_2_1_1_2_1_1_1"/><i><font color="Green">E272</font></i>: 
( <font color="Maroon">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">LC</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> )
 <b>by </b><i/>;<br/><a NAME="E12:41_1_1_1_12_2_1_1_2_1_1_1"/><i><font color="Green">E273</font></i>: 
<font color="Maroon">LC</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">LCT</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon">u</font> <a href="group_1.html#K10">*</a> <font color="Maroon">v</font></span>)</span> <a href="group_1.html#K10">*</a> <font color="Maroon">w</font></span>)</span></span>)</span></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>by </b><i><a class="ref" href="hilbasis.html#T3" target="_self">Th3</a>, , , , <a class="ref" href="finseq_2.html#T39">FINSEQ_2:39</a></i>;<br/><a NAME="E13:41_1_1_1_12_2_1_1_2_1_1_1"/><i><font color="Green">E274</font></i>: 
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
 <b>by </b><i><a class="ref" href="hilbasis.html#T3" target="_self">Th3</a>, , , , <a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>;<br/><a NAME="E14:41_1_1_1_12_2_1_1_2_1_1_1"/>
 <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font>
 <b>by </b><i>, , , , <a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>;<br/><b>hence </b><a NAME="E15:41_1_1_1_12_2_1_1_2_1_1_1"/>
 ex <font color="Olive">LC</font> being   <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font> ex <font color="Olive">sLC</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font> &amp; <font color="Olive">sLC</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">LC</font> &amp; <font color="Olive">sLC</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Olive">sLC</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font> )
 <b>by </b><i>, , <a class="ref" href="hilbasis.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_12_2_1_1_2_1_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_1_12_2_1_1_2_1_1_2"/><i><font color="Green">E275</font></i>: 
<font color="Maroon">k</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0
 ;<br/><div class="add"><a NAME="E2:41_1_1_1_12_2_1_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="xxreal_0.html#NR4" target="_self">&gt;</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:41_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="xreal_1.html#T8">XREAL_1:8</a></i>;<br/><a NAME="E3:41_1_1_1_12_2_1_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> <a href="hidden.html#NR2" target="_self">&lt;&gt;</a> 0
 ;<br/><a NAME="E4:41_1_1_1_12_2_1_1_2_1_1_2"/><b>then </b>
not <font color="Maroon">lc</font> is <a href="xboole_0.html#V1">empty</a>
 <b>by </b><i><a class="ref" href="finseq_1.html#T25">FINSEQ_1:25</a></i>;<br/><b>then consider </b><font color="Maroon">p</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>, <font color="Maroon">e</font> being    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:41_1_1_1_12_2_1_1_2_1_1_2"/><i><font color="Green">E276</font></i>: 
<font color="Maroon">lc</font> <a href="hidden.html#R1">=</a> <font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="binarith.html#K11">*&gt;</a></span>
 <b>and </b><br/><a NAME="E7:41_1_1_1_12_2_1_1_2_1_1_2"/><i><font color="Green">E277</font></i>: 
<span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="binarith.html#K11">*&gt;</a></span> is    <a href="ideal_1.html#M1">LinearCombination</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="ideal_1.html#T32">IDEAL_1:32</a></i>;<br/> <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> = 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span>
<b>by </b><i>, <a class="ref" href="finseq_1.html#T35">FINSEQ_1:35</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font></span>)</span> <a href="nat_1.html#K1">+</a> 1
<b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>
;<br/><a NAME="E9:41_1_1_1_12_2_1_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <font color="Maroon">p</font> <a href="hidden.html#R1">=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:41_1_1_1"><i><font color="Green">E59</font></i></a></i>;<br/><b>then consider </b><font color="Maroon">LCp</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font>, <font color="Maroon">sLCp</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E11:41_1_1_1_12_2_1_1_2_1_1_2"/><i><font color="Green">E278</font></i>: 
( <font color="Maroon">LCp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">p</font> &amp; <font color="Maroon">sLCp</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">LCp</font> &amp; <font color="Maroon">sLCp</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">p</font> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">sLCp</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font> )
 <b>by </b><i/>;<br/><a NAME="E12:41_1_1_1_12_2_1_1_2_1_1_2"/>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="hidden.html#R1">=</a> 0 <a href="nat_1.html#K1">+</a> 1
 <b>by </b><i><a class="ref" href="finseq_1.html#T56">FINSEQ_1:56</a></i>;<br/><a NAME="E13:41_1_1_1_12_2_1_1_2_1_1_2"/><b>then </b>
 <a href="finseq_1.html#K3">len</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="binarith.html#K11">*&gt;</a></span> <a href="xxreal_0.html#R1">&lt;=</a> <font color="Maroon">k</font>
 <b>by </b><i><a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, <a class="ref" href="nat_1.html#T38">NAT_1:38</a></i>;<br/><b>then consider </b><font color="Maroon">LCe</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font>, <font color="Maroon">sLCe</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E15:41_1_1_1_12_2_1_1_2_1_1_2"/><i><font color="Green">E279</font></i>: 
( <font color="Maroon">LCe</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Maroon">sLCe</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">LCe</font> &amp; <font color="Maroon">sLCe</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">sLCe</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font> )
 <b>by </b><i>, <a class="ref" href="hilbasis.html#T3" target="_self">Th3</a></i>;<br/><b>consider </b><font color="Maroon">LC</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font><b> such that </b><br/><a NAME="E17:41_1_1_1_12_2_1_1_2_1_1_2"/><i><font color="Green">E280</font></i>: 
( <font color="Maroon">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font> &amp;  <a href="finseq_1.html#K3">len</a> <font color="Maroon">LC</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font> )
 <b>by </b><i/>;<br/><a NAME="E18:41_1_1_1_12_2_1_1_2_1_1_2"/>
 <a href="relset_1.html#K4">dom</a> <font color="Maroon">LCT</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 <b>by </b><i><a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a></i>;<br/><a NAME="E19:41_1_1_1_12_2_1_1_2_1_1_2"/><b>then </b>
<span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <font color="Maroon">p</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1">(<span class="default"><a href="relset_1.html#K5">rng</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="tarski.html#R1">c=</a>  <a href="relset_1.html#K4">dom</a> <font color="Maroon">LCT</font>
 ;<br/><b>then consider </b><font color="Maroon">LCTp</font> being   <a href="finseq_1.html#NM1">FinSequence</a>, <font color="Maroon">LCTe</font> being   <a href="finseq_1.html#NM1">FinSequence</a><b> such that </b><br/><a NAME="E21:41_1_1_1_12_2_1_1_2_1_1_2"/><i><font color="Green">E281</font></i>: 
( <font color="Maroon">LCTp</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">p</font> &amp; <font color="Maroon">LCTe</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <span class="p1"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="binarith.html#K11">*&gt;</a></span> &amp; <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <span class="p1">(<span class="default"><font color="Maroon">p</font> <a href="finseq_1.html#K8">^</a> <span class="p2"><a href="binarith.html#K11">&lt;*</a><span class="default"><font color="Maroon">e</font></span><a href="binarith.html#K11">*&gt;</a></span></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">LCTp</font> <a href="finseq_1.html#K7">^</a> <font color="Maroon">LCTe</font> )
 <b>by </b><i/>;<br/><b>set </b><font color="Maroon">sLC</font> = <font color="Maroon">sLCp</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">sLCe</font>;<br/><i><font color="Green">E282</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">LC</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">LCp</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">LCe</font></span>)</span>
<b>by </b><i>, , , , , <a class="ref" href="rlvect_1.html#T58">RLVECT_1:58</a></i>
<br/>.= 
<font color="Maroon">sLCp</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">sLCe</font>
<b>by </b><i>, , <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>
;<br/><i><font color="Green">E283</font></i>:  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> = 
<span class="p1">(<span class="default"><a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">p</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">e</font>
<b>by </b><i>, <a class="ref" href="fvsum_1.html#T87">FVSUM_1:87</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">sLCp</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">sLCe</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i>, , <a class="ref" href="rlvect_1.html#T61">RLVECT_1:61</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">sLCp</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">sLCe</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#D6">POLYNOM3:def 6</a></i>
;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_12_2_1_1_2_1_1_2_4"><b>now </b></a><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_12_2_1_1_2_1_1_2_4_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:41_1_1_1_12_2_1_1_2_1_1_2_4_1"/>
(  <a href="algseq_1.html#K3">len</a> <font color="Maroon">sLCp</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">sLCe</font> or  <a href="algseq_1.html#K3">len</a> <font color="Maroon">sLCp</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">sLCe</font> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_12_2_1_1_2_1_1_2_4_1_1"><b>suppose </b></a><a NAME="E1:41_1_1_1_12_2_1_1_2_1_1_2_4_1_1"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">sLCp</font> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">sLCe</font>
 ;<br/><div class="add"><a NAME="E2:41_1_1_1_12_2_1_1_2_1_1_2_4_1_1"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">sLCp</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">sLCe</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">sLCe</font>
 <b>by </b><i><a class="ref" href="polynom4.html#T9">POLYNOM4:9</a></i>;<br/><b>hence </b><a NAME="E3:41_1_1_1_12_2_1_1_2_1_1_2_4_1_1"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">sLCp</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">sLCe</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="41_1_1_1_12_2_1_1_2_1_1_2_4_1_2"><b>suppose </b></a><a NAME="E1:41_1_1_1_12_2_1_1_2_1_1_2_4_1_2"/>
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">sLCp</font> <a href="xxreal_0.html#NR2" target="_self">&gt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">sLCe</font>
 ;<br/><div class="add"><a NAME="E2:41_1_1_1_12_2_1_1_2_1_1_2_4_1_2"/><b>then </b>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">sLCp</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">sLCe</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">sLCp</font>
 <b>by </b><i><a class="ref" href="polynom4.html#T9">POLYNOM4:9</a></i>;<br/><b>hence </b><a NAME="E3:41_1_1_1_12_2_1_1_2_1_1_2_4_1_2"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">sLCp</font> <a href="polynom3.html#K9">+</a> <font color="Maroon">sLCe</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T2">XXREAL_0:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div><b>hence </b><a NAME="E25:41_1_1_1_12_2_1_1_2_1_1_2"/>
 ex <font color="Olive">LC</font> being   <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font> ex <font color="Olive">sLC</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font> &amp; <font color="Olive">sLC</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">LC</font> &amp; <font color="Olive">sLC</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Olive">sLC</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font> )
 <b>by </b><i>, , <a class="ref" href="hilbasis.html#T4" target="_self">Th4</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div></div><b>end;</b></div></div><b>end;</b></div>

</div><b>end;</b></div>


</div><b>end;</b></div>
<a NAME="E3:41_1_1_1_12"/><i><font color="Green">E284</font></i>: 
for <font color="Olive">k</font> being   <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a>  holds  <font color="Maroon">S<sub>6</sub></font>[<font color="Olive">k</font>]
 
<b>from </b><i><a class="ref" href="nat_1.html#S1">NAT_1:sch 1</a>(, );<br/></i>
<b>let </b><font color="Maroon">lc</font> be    <a href="ideal_1.html#M1">LinearCombination</a> of  <a href="relset_1.html#K5">rng</a> <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="relset_1.html#K8">|</a> <span class="p2">(<span class="default"><a href="gr_cy_1.html#K1">Segm</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span>)</span>;<br/>

<b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> <b> such that </b><br/><a NAME="E5:41_1_1_1_12"/><i><font color="Green">E285</font></i>: 
<font color="Maroon">n</font> <a href="hidden.html#R1">=</a>  <a href="finseq_1.html#K3">len</a> <font color="Maroon">lc</font>
 ;<br/>
<b>thus </b><a NAME="E6:41_1_1_1_12"/>
 ex <font color="Olive">LC</font> being   <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font> ex <font color="Olive">sLC</font> being  <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font> &amp; <font color="Olive">sLC</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Olive">LC</font> &amp; <font color="Olive">sLC</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Olive">sLC</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font> )
 <b>by </b><i>, </i>;<br/>


</div><b>end;</b></div><b>then consider </b><font color="Maroon">LC</font> being    <a href="ideal_1.html#M1">LinearCombination</a> of <font color="Maroon">FM1</font>, <font color="Maroon">sLC</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E35:41_1_1_1"/><i><font color="Green">E288</font></i>: 
( <font color="Maroon">LC</font> <a href="hidden.html#R1">=</a> <font color="Maroon">LCT</font> <a href="finseqop.html#K5">*</a> <font color="Maroon">lc</font> &amp; <font color="Maroon">sLC</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">LC</font> &amp; <font color="Maroon">sLC</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K9">Sum</a> <font color="Maroon">lc</font> &amp;  <a href="algseq_1.html#K3">len</a> <font color="Maroon">sLC</font> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font> )
 ;<br/><b>set </b><font color="Maroon">f'</font> = <font color="Maroon">fm1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">sLC</font>;<br/><i><font color="Green">E334</font></i>: <span class="p1">(<span class="default"><font color="Maroon">fm1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">sLC</font></span>)</span> <a href="polynom3.html#K9">+</a> <font color="Maroon">sLC</font> = 
<font color="Maroon">fm1</font> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">sLC</font> <a href="polynom3.html#K9">+</a> <span class="p2">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">sLC</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T26">POLYNOM3:26</a></i>
<br/>.= 
<font color="Maroon">fm1</font> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><font color="Maroon">sLC</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">sLC</font></span>)</span>

<br/>.= 
<font color="Maroon">fm1</font> <a href="polynom3.html#K9">+</a> <span class="p1">(<span class="default"><a href="polynom3.html#K12">0_.</a> <font color="Maroon">R</font></span>)</span>
<b>by </b><i><a class="ref" href="polynom3.html#T30">POLYNOM3:30</a></i>
<br/>.= 
<font color="Maroon">fm1</font>
<b>by </b><i><a class="ref" href="polynom3.html#T29">POLYNOM3:29</a></i>
;<br/><a NAME="E37:41_1_1_1"/>
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><a href="polynom3.html#K10">-</a> <font color="Maroon">sLC</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font>
 <b>by </b><i>, <a class="ref" href="polynom4.html#T11">POLYNOM4:11</a></i>;<br/><a NAME="E38:41_1_1_1"/><b>then </b><i><font color="Green">E335</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fm1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">sLC</font></span>)</span> <a href="xxreal_0.html#R1">&lt;=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font>
 <b>by </b><i><a class="ref" href="polynom4.html#T9">POLYNOM4:9</a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_14"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_1_1_14"/><i><font color="Green">E336</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fm1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">sLC</font></span>)</span> <a href="hidden.html#R1">=</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font>
 ;<br/><a NAME="E2:41_1_1_1_14"/>
<span class="p1">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1 <a href="xxreal_0.html#NR2" target="_self">&gt;=</a> 0
 <b>by </b><i><a class="txt" href="hilbasis.html#E8:41_1_1_1"><i><font color="Green">E51</font></i></a></i>;<br/><b>then </b><i><font color="Green">E337</font></i>: <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> <a href="nat_1.html#K1">+</a> 1 = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="xcmplx_0.html#K6">-</a> 1</span>)</span> <a href="xcmplx_0.html#K2">+</a> 1
<b>by </b><i><a class="ref" href="binarith.html#D3">BINARITH:def 3</a></i>
<br/>.= 
 <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font>

;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">A</font> being   <a href="polynom3.html#NM1">Polynomial</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:41_1_1_1_14"/><i><font color="Green">E338</font></i>: 
( <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">a</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">A</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">A</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> )
 <b>by </b><i/>;<br/><span class="p1">(<span class="default"><font color="Maroon">fm1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">sLC</font></span>)</span> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span> = 
<span class="p1">(<span class="default"><font color="Maroon">fm1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">fm1</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font></span>)</span> <a href="binarith.html#K3">-'</a> 1</span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="hilbasis.html#E6:41_1_1_1"><i><font color="Green">E50</font></i></a>, , <a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, <a class="ref" href="polynom3.html#T27">POLYNOM3:27</a></i>
<br/>.= 
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>
;<br/><b>hence </b><a NAME="E7:41_1_1_1_14"/>
contradiction
 <b>by </b><i>, <a class="txt" href="hilbasis.html#E10:41_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="algseq_1.html#T25">ALGSEQ_1:25</a></i>;<br/></div><b>end;</b></div><a NAME="E40:41_1_1_1"/><b>then </b><i><font color="Green">E339</font></i>: 
 <a href="algseq_1.html#K3">len</a> <span class="p1">(<span class="default"><font color="Maroon">fm1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">sLC</font></span>)</span> <a href="xxreal_0.html#NR3" target="_self">&lt;</a>  <a href="algseq_1.html#K3">len</a> <font color="Maroon">fm1</font>
 <b>by </b><i>, <a class="ref" href="xxreal_0.html#T1">XXREAL_0:1</a></i>;<br/><a NAME="E41:41_1_1_1"/><i><font color="Green">E340</font></i>: 
<font color="Maroon">sLC</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FM1</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T60">IDEAL_1:60</a></i>;<br/><b>consider </b><font color="Maroon">n</font> being    <a href="subset_1.html#M2">Element</a> of  <a href="numbers.html#K5">NAT</a> , <font color="Maroon">A</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span>, <font color="Maroon">B</font> being  non <a href="xboole_0.html#V1">empty</a> <a href="subset_1.html#NM2">Subset</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span><b> such that </b><br/><a NAME="E43:41_1_1_1"/><i><font color="Green">E341</font></i>: 
( <font color="Maroon">A</font> <a href="hidden.html#R1">=</a> <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font> &amp; <font color="Maroon">B</font> <a href="hidden.html#R1">=</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">A</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span> &amp; <font color="Maroon">n</font> <a href="hidden.html#R1">=</a> <font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1 &amp; <font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">n</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">F</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">n</font></span>)</span> <a href="subset_1.html#K4">\/</a> <span class="p1"><a href="domain_1.html#K6">{</a><span class="default"><span class="p2">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p3">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span></span><a href="domain_1.html#K6">}</a></span> &amp; <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a>  <a href="hilbasis.html#K4" target="_self">minlen</a> <font color="Maroon">B</font> )
 <b>by </b><i><a class="txt" href="hilbasis.html#E3:41_1_1_1"><i><font color="Green">E44</font></i></a></i>;<br/><div><i><font color="Green">E342</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_15"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:41_1_1_1_15"/><i><font color="Green">E343</font></i>: 
<font color="Maroon">fm1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">sLC</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FM1</font> <a href="ideal_1.html#K7">-Ideal</a> 
 ;<br/><b>reconsider </b><font color="Maroon">sLC</font> = <font color="Maroon">sLC</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">fm1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">sLC</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E4:41_1_1_1_15"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">f'</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">sLC</font>
 <b>by </b><i>, <a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/><a NAME="E5:41_1_1_1_15"/><b>then </b>
<font color="Maroon">fm1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">FM1</font> <a href="ideal_1.html#K7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:41_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="hilbasis.html#T3" target="_self">Th3</a>, <a class="ref" href="ideal_1.html#D1">IDEAL_1:def 1</a></i>;<br/><b>hence </b><a NAME="E6:41_1_1_1_15"/>
contradiction
 <b>by </b><i><a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/></div><b>end;</b></div><a NAME="E45:41_1_1_1"/>
<font color="Maroon">fm1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">sLC</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="41_1_1_1_16"><b>proof </b></a><div class="add">

<a NAME="E1:41_1_1_1_16"/>
<font color="Maroon">FM1</font> is   <a href="subset_1.html#NM2">Subset</a> of <font color="Maroon">I</font>
 
<b>by </b><i/>;<br/>
<a NAME="E2:41_1_1_1_16"/><b>then </b>
<font color="Maroon">FM1</font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font> <a href="ideal_1.html#K7">-Ideal</a> 
 
<b>by </b><i><a class="ref" href="ideal_1.html#T57">IDEAL_1:57</a></i>;<br/>
<a NAME="E3:41_1_1_1_16"/><b>then </b><i><font color="Green">E344</font></i>: 
<font color="Maroon">FM1</font> <a href="ideal_1.html#K7">-Ideal</a>  <a href="tarski.html#R1">c=</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#T44">IDEAL_1:44</a></i>;<br/>
<b>reconsider </b><font color="Maroon">sLC</font> = <font color="Maroon">sLC</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<b>reconsider </b><font color="Maroon">f'</font> = <font color="Maroon">fm1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">sLC</font> as    <a href="subset_1.html#M1">Element</a> of the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font></span>)</span> <b>by </b><i><a class="ref" href="polynom3.html#D12">POLYNOM3:def 12</a></i>;<br/>
<a NAME="E6:41_1_1_1_16"/><i><font color="Green">E345</font></i>: 
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p1">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/>
<a NAME="E7:41_1_1_1_16"/>
<font color="Maroon">f'</font> <a href="hidden.html#R1">=</a> <span class="p1">(<span class="default"><font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <span class="p2">(<span class="default"><font color="Maroon">m</font> <a href="nat_1.html#K1">+</a> 1</span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">sLC</font>
 
<b>by </b><i/>;<br/>
<b>hence </b><a NAME="E8:41_1_1_1_16"/>
<font color="Maroon">fm1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">sLC</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 <b>by </b><i><a class="txt" href="hilbasis.html#E10:41_1_1_1"><i><font color="Green">E59</font></i></a>, <a class="ref" href="hilbasis.html#T3" target="_self">Th3</a>, , <a class="ref" href="ideal_1.html#T15">IDEAL_1:15</a></i>;<br/>


</div><b>end;</b></div><a NAME="E46:41_1_1_1"/><b>then </b>
<font color="Maroon">fm1</font> <a href="polynom3.html#K11">-</a> <font color="Maroon">sLC</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> <a href="subset_1.html#K7">\</a> <span class="p1">(<span class="default"><font color="Maroon">FM1</font> <a href="ideal_1.html#K7">-Ideal</a> </span>)</span>
 <b>by </b><i>, <a class="ref" href="xboole_0.html#D4">XBOOLE_0:def 4</a></i>;<br/><b>hence </b><a NAME="E47:41_1_1_1"/>
contradiction
 <b>by </b><i>, <a class="ref" href="hilbasis.html#T2" target="_self">Th2</a>, </i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:41_1_1"/>
 <a href="polynom3.html#K16">Polynom-Ring</a> <font color="Maroon">R</font> is <a href="ideal_1.html#V7">Noetherian</a>
 ;<br/>


</div>
