<?xml version="1.0"?>
<div class="add">

<b>let </b><font color="Maroon" title="c1">R</font> be  non <a href="struct_0.html#V2" title="STRUCT_0:attr.2">empty</a> non <a href="struct_0.html#V7" title="STRUCT_0:attr.7">trivial</a> <a href="algstr_0.html#V13" title="ALGSTR_0:attr.13">right_complementable</a> <a href="rlvect_1.html#V2" title="RLVECT_1:attr.2">Abelian</a> <a href="rlvect_1.html#V3" title="RLVECT_1:attr.3">add-associative</a> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">right_zeroed</a> <a href="group_1.html#V3" title="GROUP_1:attr.3">associative</a> <a href="group_1.html#V5" title="GROUP_1:attr.5">commutative</a> <a href="vectsp_1.html#V4" title="VECTSP_1:attr.4">well-unital</a> <a href="vectsp_1.html#V5" title="VECTSP_1:attr.5">distributive</a>  <a href="algstr_0.html#L6" title="ALGSTR_0:struct.6">doubleLoopStr</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> for <font color="Olive" title="b1">X</font> being <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>  holds not  <a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Olive" title="b1">X</font>,<font color="Maroon" title="c1">R</font> is <a href="ideal_1.html#V6" title="IDEAL_1:attr.6">Noetherian</a></span><br/><b>let </b><font color="Maroon" title="c2">X</font> be  <a href="finset_1.html#NV2" title="FINSET_1:NV.2">infinite</a> <a href="ordinal1.html#NM2" title="ORDINAL1:NM.2">Ordinal</a>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> not  <a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font> is <a href="ideal_1.html#V6" title="IDEAL_1:attr.6">Noetherian</a></span><br/>



<b>assume </b><a NAME="E1:45"/><i><font color="Green" title="E38">A1</font></i>: 
 <a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font> is <a href="ideal_1.html#V6" title="IDEAL_1:attr.6">Noetherian</a>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/>

<b>set </b><font color="Maroon" title="c3">tcR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c4">tcPR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font></span>)</span>;<br/>
<a NAME="E2:45"/><i><font color="Green" title="E39">A2</font></i>: 
 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <a href="ordinal1.html#R1" title="ORDINAL1:pred.1">c=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="card_3.html#T102" title="CARD_3:th.102">CARD_3:102</a></i>;<br/>
<b>set </b><font color="Maroon" title="c5">S</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  </span>}</span> ;<br/>
<a NAME="E3:45"/><i><font color="Green" title="E40">A3</font></i>: 
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c6">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  </span>}</span>  or <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font></span>)</span> )</span><br/>

<b>assume </b><a NAME="E1:45_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font></span>)</span></span><br/>

<b>then consider </b><font color="Maroon" title="c7">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:45_1"/><i><font color="Green" title="E41">A4</font></i>: 
( <font color="Maroon" title="c6">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c7">n</font>,<font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c7">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 ;<br/>
<b>thus </b><a NAME="E4:45_1"/>
<font color="Maroon" title="c6">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E3:45_1"><i><font color="Green" title="E41">A4</font></i></a>, <a class="ref" href="polynom1.html#D27" title="POLYNOM1:def.27">POLYNOM1:def 27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<a NAME="E4:45"/>
 <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c6">0X</font> =  <a href="numbers.html#K6" title="NUMBERS:func.6">0</a>  as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="#E2:45"><i><font color="Green" title="E39">A2</font></i></a></i>;<br/>
<a NAME="E6:45"/>
 <a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c6">0X</font>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  </span>}</span> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c7">S</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Olive" title="b1">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Olive" title="b1">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font></span>)</span> <b>by </b><i><a class="txt" href="#E3:45"><i><font color="Green" title="E40">A3</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">C</font> being  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p1">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font></span>)</span><b> such that </b><br/><a NAME="E9:45"/><i><font color="Green" title="E41">A5</font></i>: 
<font color="Maroon" title="c8">C</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">S</font>
 <b>and </b><br/><a NAME="E10:45"/><i><font color="Green" title="E42">A6</font></i>: 
<font color="Maroon" title="c8">C</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c7">S</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="#E1:45"><i><font color="Green" title="E38">A1</font></i></a>, <a class="ref" href="ideal_1.html#T99" title="IDEAL_1:th.99">IDEAL_1:99</a></i>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>1</sub></font>(   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ) <b>-&gt; </b>   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>  = $1;<br/>
<b>deffunc </b><font color="Maroon">H<sub>2</sub></font>(   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>) <b>-&gt; </b>  <a href="polynom1.html#NM3" title="POLYNOM1:NM.3">Series</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font> =  <a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> $1,<font color="Maroon" title="c1">R</font>;<br/>
<b>set </b><font color="Maroon" title="c9">CN</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font> </span>}</span> ;<br/>
<a NAME="E11:45"/><i><font color="Green" title="E43">A7</font></i>: 
<font color="Maroon" title="c8">C</font> is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
;<br/>
<a NAME="E12:45"/><i><font color="Green" title="E44">A8</font></i>: 
for <font color="Olive" title="b1">d1</font>, <font color="Olive" title="b2">d2</font> being   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font>  st <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">d1</font>) <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a> <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b2">d2</font>) holds <br/><font color="Olive" title="b1">d1</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">d2</font>
 
<b>by </b><i><a class="ref" href="hilbasis.html#T14" target="_self" title="HILBASIS:th.14">Th14</a></i>;<br/>
<a NAME="E13:45"/><i><font color="Green" title="E45">A9</font></i>: 
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font> </span>}</span>  is <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a>
 
<b>from </b><i><a class="ref" href="funct_7.html#S2" title="FUNCT_7:sch.2">FUNCT_7:sch 2</a>(<a class="txt" href="#E11:45"><i><font color="Green" title="E43">A7</font></i></a>, <a class="txt" href="#E12:45"><i><font color="Green" title="E44">A8</font></i></a>);<br/></i>
<b>consider </b><font color="Maroon" title="c10">c</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">C</font>;<br/>
<a NAME="E15:45"/>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font>
 
;<br/>
<a NAME="E16:45"/><b>then </b>
<font color="Maroon" title="c10">c</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="txt" href="#E9:45"><i><font color="Green" title="E41">A5</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c11">cn</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E18:45"/><i><font color="Green" title="E46">A10</font></i>: 
( <font color="Maroon" title="c10">c</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c11">cn</font>,<font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c11">cn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 ;<br/>
<b>reconsider </b><font color="Maroon" title="c12">cn</font> = <font color="Maroon" title="c11">cn</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E18:45"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/>
<a NAME="E20:45"/><i><font color="Green" title="E47">A11</font></i>: 
<font color="Maroon" title="c12">cn</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="#E18:45"><i><font color="Green" title="E46">A10</font></i></a></i>;<br/>
<a NAME="E21:45"/>
<span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font> </span>}</span>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c13">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <i><font color="Red">:: according to </font></i><a class="ref" href="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a> <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> (  not <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font> </span>}</span>  or <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )</span><br/>

<b>assume </b><a NAME="E1:45_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font> </span>}</span> 
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> </span><br/>

<b>then consider </b><font color="Maroon" title="c14">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E3:45_2"/><i><font color="Green" title="E48">A12</font></i>: 
( <font color="Maroon" title="c13">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c14">n</font> &amp;  <a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c14">n</font>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font> )
 ;<br/>
<a NAME="E4:45_2"/>
 <a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c14">n</font>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 
<b>by </b><i><a class="txt" href="#E9:45"><i><font color="Green" title="E41">A5</font></i></a>, <a class="txt" href="#E3:45_2"><i><font color="Green" title="E48">A12</font></i></a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c15">m</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E6:45_2"/><i><font color="Green" title="E49">A13</font></i>: 
(  <a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c14">n</font>,<font color="Maroon" title="c1">R</font> <a href="funct_2.html#R2" title="FUNCT_2:pred.2">=</a>  <a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c15">m</font>,<font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c15">m</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 ;<br/>
<b>thus </b><a NAME="E7:45_2"/>
<font color="Maroon" title="c13">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 <b>by </b><i><a class="txt" href="#E3:45_2"><i><font color="Green" title="E48">A12</font></i></a>, <a class="txt" href="#E6:45_2"><i><font color="Green" title="E49">A13</font></i></a>, <a class="ref" href="hilbasis.html#T14" target="_self" title="HILBASIS:th.14">Th14</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c13">CN</font> = <span class="p1">{<span class="default"> <font color="Maroon">H<sub>1</sub></font>(<font color="Olive" title="b1">n</font>) where <font color="Olive" title="b1">n</font> is    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> : <font color="Maroon">H<sub>2</sub></font>(<font color="Olive" title="b1">n</font>) <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="finset_1.html#V1" title="FINSET_1:attr.1">finite</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="#E13:45"><i><font color="Green" title="E45">A9</font></i></a>, <a class="txt" href="#E20:45"><i><font color="Green" title="E47">A11</font></i></a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c14">mm</font> =  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c13">CN</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="ref" href="ordinal1.html#D13" title="ORDINAL1:def.13">ORDINAL1:def 13</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c15">m1</font> = <font color="Maroon" title="c14">mm</font> <a href="nat_1.html#K2" title="NAT_1:func.2">+</a> 1 as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/>
<a NAME="E25:45"/>
<font color="Maroon" title="c15">m1</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> 
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c16">m2</font> = <font color="Maroon" title="c15">m1</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font> <b>by </b><i><a class="txt" href="#E2:45"><i><font color="Green" title="E39">A2</font></i></a></i>;<br/>
<a NAME="E27:45"/>
(  <a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c16">m2</font>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font> &amp; <font color="Maroon" title="c7">S</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c7">S</font> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  )
 
<b>by </b><i><a class="ref" href="ideal_1.html#D15" title="IDEAL_1:def.15">IDEAL_1:def 15</a></i>;<br/>
<b>then consider </b><font color="Maroon" title="c17">lc</font> being    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c8">C</font><b> such that </b><br/><a NAME="E29:45"/><i><font color="Green" title="E48">A14</font></i>: 
 <a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c16">m2</font>,<font color="Maroon" title="c1">R</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c17">lc</font>
 <b>by </b><i><a class="txt" href="#E10:45"><i><font color="Green" title="E42">A6</font></i></a>, <a class="ref" href="ideal_1.html#T60" title="IDEAL_1:th.60">IDEAL_1:60</a></i>;<br/>
<b>reconsider </b><font color="Maroon" title="c18">f0</font> = <font color="Maroon" title="c2">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> as   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of <font color="Maroon" title="c2">X</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>reconsider </b><font color="Maroon" title="c19">ev</font> = <font color="Maroon" title="c18">f0</font> <a href="funct_7.html#K15" title="FUNCT_7:func.15">+*</a> <font color="Maroon" title="c16">m2</font>,<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">R</font></span>)</span> as   <a href="struct_0.html#NM5" title="STRUCT_0:NM.5">Function</a> of <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font> ;<br/>
<a NAME="E32:45"/>
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c2">X</font>
 
<b>by </b><i><a class="ref" href="funcop_1.html#T19" title="FUNCOP_1:th.19">FUNCOP_1:19</a></i>;<br/>
<a NAME="E33:45"/><b>then </b><i><font color="Green" title="E49">A15</font></i>: 
<font color="Maroon" title="c19">ev</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">m2</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">R</font>
 
<b>by </b><i><a class="ref" href="funct_7.html#T33" title="FUNCT_7:th.33">FUNCT_7:33</a></i>;<br/>
<a NAME="E34:45"/><i><font color="Green" title="E50">A16</font></i>: 
 <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c16">m2</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c16">m2</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 
<b>by </b><i><a class="ref" href="hilbasis.html#T13" target="_self" title="HILBASIS:th.13">Th13</a></i>;<br/>
<i><font color="Green" title="E51">A17</font></i>: <span class="p1">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c16">m2</font>,<font color="Maroon" title="c1">R</font></span>)</span> = 
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c16">m2</font>,<font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c19">ev</font>
<b>by </b><i><a class="ref" href="polynom2.html#D5" title="POLYNOM2:def.5">POLYNOM2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c16">m2</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c16">m2</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c16">m2</font></span>)</span>,<font color="Maroon" title="c19">ev</font></span>)</span>
<b>by </b><i><a class="txt" href="#E34:45"><i><font color="Green" title="E50">A16</font></i></a>, <a class="ref" href="polynom2.html#T21" title="POLYNOM2:th.21">POLYNOM2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c16">m2</font></span>)</span>,<font color="Maroon" title="c19">ev</font></span>)</span>
<b>by </b><i><a class="ref" href="hilbasis.html#T12" target="_self" title="HILBASIS:th.12">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">ev</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c16">m2</font></span>)</span>
<b>by </b><i><a class="ref" href="hilbasis.html#T11" target="_self" title="HILBASIS:th.11">Th11</a></i>
<br/>.= 
 <a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="txt" href="#E33:45"><i><font color="Green" title="E49">A15</font></i></a>, <a class="ref" href="group_1.html#D5" title="GROUP_1:def.5">GROUP_1:def 5</a></i>
;<br/>
<a NAME="E36:45"/>
the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>
 
;<br/>
<b>then reconsider </b><font color="Maroon" title="c20">cR</font> = the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="subset_1.html#NM2" title="SUBSET_1:NM.2">Subset</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> ;<br/>
<b>consider </b><font color="Maroon" title="c21">E</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of <span class="p1"><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">[:</a><span class="default">the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font></span>)</span>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <span class="p2">(<span class="default"><a href="polynom1.html#K28" title="POLYNOM1:func.28">Polynom-Ring</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font></span>)</span></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span><b> such that </b><br/><a NAME="E39:45"/><i><font color="Green" title="E52">A18</font></i>: 
<font color="Maroon" title="c21">E</font> <a href="ideal_1.html#R1" title="IDEAL_1:pred.1">represents</a> <font color="Maroon" title="c17">lc</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T35" title="IDEAL_1:th.35">IDEAL_1:35</a></i>;<br/>
<b>set </b><font color="Maroon" title="c22">P</font> =  <a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font>;<br/>
<b>deffunc </b><font color="Maroon">H<sub>3</sub></font>(  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>) <b>-&gt; </b>   <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font> = <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1</span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></span>)</span>;<br/>
<b>consider </b><font color="Maroon" title="c23">LC</font> being    <a href="finseq_1.html#M2" title="FINSEQ_1:mode.2">FinSequence</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E41:45"/><i><font color="Green" title="E53">A19</font></i>: 
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c23">LC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">lc</font>
 <b>and </b><br/><a NAME="E42:45"/><i><font color="Green" title="E54">A20</font></i>: 
for <font color="Olive" title="b1">k</font> being  <a href="nat_1.html#NM1" title="NAT_1:NM.1">Nat</a>  st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">LC</font> holds <br/><font color="Maroon" title="c23">LC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon">H<sub>3</sub></font>(<font color="Olive" title="b1">k</font>)
 <b>from </b><i><a class="ref" href="finseq_2.html#S1" title="FINSEQ_2:sch.1">FINSEQ_2:sch 1</a>();<br/></i>
<a NAME="E43:45"/><i><font color="Green" title="E55">A21</font></i>: 
 <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">LC</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">lc</font></span>)</span>
 
<b>by </b><i><a class="txt" href="#E41:45"><i><font color="Green" title="E53">A19</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_4"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c24">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">LC</font> implies  ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b3">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c20">cR</font> st <font color="Maroon" title="c23">LC</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">v</font> )</span><br/><b>assume </b><a NAME="E1:45_4"/><i><font color="Green" title="E56">A22</font></i>: 
<font color="Maroon" title="c24">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c23">LC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">u</font>, <font color="Olive" title="b2">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b3">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c20">cR</font> st <font color="Maroon" title="c23">LC</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b3">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">v</font></span><br/><b>then reconsider </b><font color="Maroon" title="c25">k</font> = <font color="Maroon" title="c24">i</font> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  ;<br/><a NAME="E3:45_4"/><i><font color="Green" title="E57">A23</font></i>: 
<font color="Maroon" title="c25">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">lc</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E41:45"><i><font color="Green" title="E53">A19</font></i></a>, <a class="txt" href="#E1:45_4"><i><font color="Green" title="E56">A22</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">u</font> = <span class="p1">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">k</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span>, <font color="Maroon" title="c27">v</font> = <span class="p1">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">k</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/><b>reconsider </b><font color="Maroon" title="c28">a</font> = <span class="p1">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">k</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span> as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c20">cR</font> ;<br/><b>take </b><font color="Maroon" title="c29">u</font> = <font color="Maroon" title="c26">u</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">v</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ex <font color="Olive" title="b2">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c20">cR</font> st <font color="Maroon" title="c23">LC</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b2">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">v</font></span><br/><b>take </b><font color="Maroon" title="c30">v</font> = <font color="Maroon" title="c27">v</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide">  ex <font color="Olive" title="b1">a</font> being   <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c20">cR</font> st <font color="Maroon" title="c23">LC</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Olive" title="b1">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c30">v</font></span><br/><b>take </b><font color="Maroon" title="c31">a</font> = <font color="Maroon" title="c28">a</font>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c23">LC</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c29">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c31">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c30">v</font></span><br/><b>thus </b><font color="Maroon" title="c23">LC</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c24">i</font> = 
<font color="Maroon" title="c23">LC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">k</font>
<b>by </b><i><a class="txt" href="#E1:45_4"><i><font color="Green" title="E56">A22</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon" title="c29">u</font> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c31">a</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <font color="Maroon" title="c30">v</font>
<b>by </b><i><a class="txt" href="#E42:45"><i><font color="Green" title="E54">A20</font></i></a>, <a class="txt" href="#E3:45_4"><i><font color="Green" title="E57">A23</font></i></a>, <a class="txt" href="#E43:45"><i><font color="Green" title="E55">A21</font></i></a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon" title="c24">LC</font> = <font color="Maroon" title="c23">LC</font> as    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c20">cR</font> <b>by </b><i><a class="ref" href="ideal_1.html#D9" title="IDEAL_1:def.9">IDEAL_1:def 9</a></i>;<br/>
<div><i><font color="Green" title="E56">A24</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c25">i</font> be    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> ( <font color="Maroon" title="c25">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">LC</font> implies <font color="Maroon" title="c24">LC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )</span><br/><b>assume </b><a NAME="E1:45_5"/><i><font color="Green" title="E57">A25</font></i>: 
<font color="Maroon" title="c25">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">LC</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> <font color="Maroon" title="c24">LC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span><br/><a NAME="E2:45_5"/><b>then </b><i><font color="Green" title="E58">A26</font></i>: 
<font color="Maroon" title="c25">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c17">lc</font>
 <b>by </b><i><a class="txt" href="#E41:45"><i><font color="Green" title="E53">A19</font></i></a>, <a class="ref" href="finseq_3.html#T31" title="FINSEQ_3:th.31">FINSEQ_3:31</a></i>;<br/><a NAME="E3:45_5"/><i><font color="Green" title="E59">A27</font></i>: 
<font color="Maroon" title="c25">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K2" title="FINSEQ_1:func.2">Seg</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c17">lc</font></span>)</span>
 <b>by </b><i><a class="txt" href="#E41:45"><i><font color="Green" title="E53">A19</font></i></a>, <a class="txt" href="#E1:45_5"><i><font color="Green" title="E57">A25</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c26">y</font> = <span class="p1">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  as    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c8">C</font> <b>by </b><i><a class="txt" href="#E39:45"><i><font color="Green" title="E52">A18</font></i></a>, <a class="txt" href="#E2:45_5"><i><font color="Green" title="E58">A26</font></i></a>, <a class="ref" href="ideal_1.html#D12" title="IDEAL_1:def.12">IDEAL_1:def 12</a></i>;<br/><a NAME="E5:45_5"/>
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c8">C</font>
 ;<br/><a NAME="E6:45_5"/><b>then </b>
<font color="Maroon" title="c26">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c7">S</font>
 <b>by </b><i><a class="txt" href="#E9:45"><i><font color="Green" title="E41">A5</font></i></a></i>;<br/><b>then consider </b><font color="Maroon" title="c27">n</font> being    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</a> of <font color="Maroon" title="c2">X</font><b> such that </b><br/><a NAME="E8:45_5"/><i><font color="Green" title="E60">A28</font></i>: 
( <font color="Maroon" title="c26">y</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c27">n</font>,<font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c27">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  )
 ;<br/><a NAME="E9:45_5"/><i><font color="Green" title="E61">A29</font></i>: 
<font color="Maroon" title="c27">n</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">CN</font>
 <b>by </b><i><a class="txt" href="#E8:45_5"><i><font color="Green" title="E60">A28</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="45_5_1"><b>now </b></a><div class="add"><b>assume </b><a NAME="E1:45_5_1"/>
<font color="Maroon" title="c16">m2</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c13">CN</font>
 ; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> contradiction</span><br/><a NAME="E2:45_5_1"/><b>then </b>
<span class="p1">(<span class="default"><a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c13">CN</font></span>)</span> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="xxreal_2.html#K1" title="XXREAL_2:func.1">max</a> <font color="Maroon" title="c13">CN</font>
 <b>by </b><i><a class="ref" href="xxreal_2.html#D8" title="XXREAL_2:def.8">XXREAL_2:def 8</a></i>;<br/><b>hence </b><a NAME="E3:45_5_1"/>
contradiction
 <b>by </b><i><a class="ref" href="xreal_1.html#T31" title="XREAL_1:th.31">XREAL_1:31</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div><b>then </b><i><font color="Green" title="E62">A30</font></i>: <font color="Maroon" title="c19">ev</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">n</font> = 
<span class="p1">(<span class="default"><font color="Maroon" title="c2">X</font> <a href="funcop_1.html#K8" title="FUNCOP_1:func.8">--&gt;</a> <span class="p2">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">n</font>
<b>by </b><i><a class="txt" href="#E9:45_5"><i><font color="Green" title="E61">A29</font></i></a>, <a class="ref" href="funct_7.html#T34" title="FUNCT_7:th.34">FUNCT_7:34</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="ref" href="funcop_1.html#T13" title="FUNCOP_1:th.13">FUNCOP_1:13</a></i>
;<br/><a NAME="E12:45_5"/><i><font color="Green" title="E63">A31</font></i>: 
 <a href="polynom1.html#K10" title="POLYNOM1:func.10">Support</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c27">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c27">n</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="hilbasis.html#T13" target="_self" title="HILBASIS:th.13">Th13</a></i>;<br/><i><font color="Green" title="E64">A32</font></i>: <span class="p1">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c27">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> = 
 <a href="polynom2.html#K5" title="POLYNOM2:func.5">eval</a> <span class="p1">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c27">n</font>,<font color="Maroon" title="c1">R</font></span>)</span>,<font color="Maroon" title="c19">ev</font>
<b>by </b><i><a class="ref" href="polynom2.html#D5" title="POLYNOM2:def.5">POLYNOM2:def 5</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="hilbasis.html#K3" title="HILBASIS:func.3">1_1</a> <font color="Maroon" title="c27">n</font>,<font color="Maroon" title="c1">R</font></span>)</span> <a href="polynom1.html#K13" title="POLYNOM1:func.13">.</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c27">n</font></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c27">n</font></span>)</span>,<font color="Maroon" title="c19">ev</font></span>)</span>
<b>by </b><i><a class="txt" href="#E12:45_5"><i><font color="Green" title="E63">A31</font></i></a>, <a class="ref" href="polynom2.html#T21" title="POLYNOM2:th.21">POLYNOM2:21</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><a href="polynom2.html#K3" title="POLYNOM2:func.3">eval</a> <span class="p2">(<span class="default"><a href="hilbasis.html#K2" title="HILBASIS:func.2">UnitBag</a> <font color="Maroon" title="c27">n</font></span>)</span>,<font color="Maroon" title="c19">ev</font></span>)</span>
<b>by </b><i><a class="ref" href="hilbasis.html#T12" target="_self" title="HILBASIS:th.12">Th12</a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="group_1.html#K1" title="GROUP_1:func.1">1_</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><font color="Maroon" title="c19">ev</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c27">n</font></span>)</span>
<b>by </b><i><a class="ref" href="hilbasis.html#T11" target="_self" title="HILBASIS:th.11">Th11</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="txt" href="#E11:45_5"><i><font color="Green" title="E62">A30</font></i></a>, <a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
;<br/><b>thus </b><font color="Maroon" title="c24">LC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c25">i</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E42:45"><i><font color="Green" title="E54">A20</font></i></a>, <a class="txt" href="#E3:45_5"><i><font color="Green" title="E59">A27</font></i></a>, <a class="txt" href="#E43:45"><i><font color="Green" title="E55">A21</font></i></a></i>
<br/>.= 
<span class="p1">(<span class="default"><a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c25">i</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></span>)</span>
<b>by </b><i><a class="txt" href="#E8:45_5"><i><font color="Green" title="E60">A28</font></i></a>, <a class="txt" href="#E13:45_5"><i><font color="Green" title="E64">A32</font></i></a>, <a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="ref" href="vectsp_1.html#T36" title="VECTSP_1:th.36">VECTSP_1:36</a></i>
; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/></div><b>end;</b></div>
<a NAME="E47:45"/>
for <font color="Olive" title="b1">k</font> being   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a>   st <font color="Olive" title="b1">k</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="finseq_1.html#K4" title="FINSEQ_1:func.4">dom</a> <font color="Maroon" title="c24">LC</font> holds <br/><font color="Maroon" title="c24">LC</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p3">(<span class="default"><span class="p4">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span></span>)</span> <a href="group_1.html#K8" title="GROUP_1:func.8">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c21">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="#E42:45"><i><font color="Green" title="E54">A20</font></i></a></i>;<br/>
<b>then </b><span class="p1">(<span class="default"><a href="polynom2.html#K6" title="POLYNOM2:func.6">Polynom-Evaluation</a> <font color="Maroon" title="c2">X</font>,<font color="Maroon" title="c1">R</font>,<font color="Maroon" title="c19">ev</font></span>)</span> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c17">lc</font></span>)</span> = 
 <a href="rlvect_1.html#K7" title="RLVECT_1:func.7">Sum</a> <font color="Maroon" title="c24">LC</font>
<b>by </b><i><a class="txt" href="#E39:45"><i><font color="Green" title="E52">A18</font></i></a>, <a class="txt" href="#E41:45"><i><font color="Green" title="E53">A19</font></i></a>, <a class="ref" href="hilbasis.html#T24" target="_self" title="HILBASIS:th.24">Th24</a></i>
<br/>.= 
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
<b>by </b><i><a class="txt" href="#E46:45"><i><font color="Green" title="E56">A24</font></i></a>, <a class="ref" href="polynom3.html#T1" title="POLYNOM3:th.1">POLYNOM3:1</a></i>
;<br/>
<b>hence </b><a NAME="E49:45"/>
contradiction
 <b>by </b><i><a class="txt" href="#E29:45"><i><font color="Green" title="E48">A14</font></i></a>, <a class="txt" href="#E35:45"><i><font color="Green" title="E51">A17</font></i></a>, <a class="ref" href="polynom1.html#T27" title="POLYNOM1:th.27">POLYNOM1:27</a></i>; <a class="txt" onclick="hs(this)" href="javascript:()"><i><font color="Red">::  thesis: </font></i></a><span class="hide"> verum</span><br/>


</div>
