<?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> <a href="rlvect_1.html#V4" title="RLVECT_1:attr.4">add-associative</a> <a href="rlvect_1.html#V5" title="RLVECT_1:attr.5">right_zeroed</a> <a href="rlvect_1.html#V6" title="RLVECT_1:attr.6">right_complementable</a> <a href="vectsp_1.html#V7" title="VECTSP_1:attr.7">well-unital</a> <a href="vectsp_1.html#V8" title="VECTSP_1:attr.8">distributive</a> <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a> <a href="int_3.html#V1" title="INT_3:attr.1">Euclidian</a>  <a href="vectsp_1.html#L3" title="VECTSP_1:struct.3">doubleLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">I</font> be   <a href="ideal_1.html#NM1" title="IDEAL_1:NM.1">Ideal</a> of <font color="Maroon" title="c1">R</font>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D29" target="_self" title="IDEAL_1:def.29">IDEAL_1:def 29</a><br/>



<div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="161_1"><b>per </b></a><a class="txt" onclick="hs(this)" href="javascript:()"><b>cases </b></a><span class="hide"><a NAME="E1:161_1"/>
( <font color="Maroon" title="c2">I</font> <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> or <font color="Maroon" title="c2">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> )
 </span>;<br/><div class="add"><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="161_1_1"><b>suppose </b></a><a NAME="E1:161_1_1"/><i><font color="Green" title="E65">A1</font></i>: 
<font color="Maroon" title="c2">I</font> <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><b>set </b><font color="Maroon" title="c3">e</font> =  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>;<br/><b>take </b>
 <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D28" target="_self" title="IDEAL_1:def.28">IDEAL_1:def 28</a><br/><b>thus </b><a NAME="E2:161_1_1"/>
<font color="Maroon" title="c2">I</font> <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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:161_1_1"><i><font color="Green" title="E65">A1</font></i></a>, <a class="ref" href="ideal_1.html#T44" target="_self" title="IDEAL_1:th.44">Th44</a></i>;<br/></div><b>end;</b></div><div><a class="txt" onclick="hsNdiv(this)" href="javascript:()" title="161_1_2"><b>suppose </b></a><a NAME="E1:161_1_2"/>
<font color="Maroon" title="c2">I</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</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="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font></span>)</span></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 ;<br/><div class="add"><b>then consider </b><font color="Maroon" title="c3">x</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E3:161_1_2"/><i><font color="Green" title="E65">A2</font></i>: 
( <font color="Maroon" title="c3">x</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> &amp; <font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> )
 <b>by </b><i><a class="ref" href="ideal_1.html#T96" target="_self" title="IDEAL_1:th.96">Th96</a></i>;<br/><b>set </b><font color="Maroon" title="c4">I'</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> </span>}</span> ;<br/><a NAME="E4:161_1_2"/><i><font color="Green" title="E66">A3</font></i>: 
<font color="Maroon" title="c3">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> </span>}</span> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:161_1_2"><i><font color="Green" title="E65">A2</font></i></a></i>;<br/><a NAME="E5:161_1_2"/>
<span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> </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 <font color="Maroon" title="c1">R</font>
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="161_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c5">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><br/>

<b>assume </b><a NAME="E1:161_1_2_1"/>
<font color="Maroon" title="c5">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c6">y</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font><b> such that </b><br/><a NAME="E3:161_1_2_1"/><i><font color="Green" title="E67">A4</font></i>: 
( <font color="Maroon" title="c5">x</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">y</font> &amp; <font color="Maroon" title="c6">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 ;<br/>
<b>thus </b><a NAME="E4:161_1_2_1"/>
<font color="Maroon" title="c5">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 <font color="Maroon" title="c1">R</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:161_1_2_1"><i><font color="Green" title="E67">A4</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c5">I'</font> = <span class="p1">{<span class="default"> <font color="Olive" title="b1">y</font> where <font color="Olive" title="b1">y</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font> : <font color="Olive" title="b1">y</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> </span>}</span>  as  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font> <b>by </b><i><a class="txt" href="ideal_1.html#E4:161_1_2"><i><font color="Green" title="E66">A3</font></i></a></i>;<br/><b>consider </b><font color="Maroon" title="c6">f</font> being   <a href="funct_2.html#NM1" title="FUNCT_2:NM.1">Function</a> of the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>, <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> <b> such that </b><br/><a NAME="E8:161_1_2"/><i><font color="Green" title="E67">A5</font></i>: 
for <font color="Olive" title="b1">a</font>, <font color="Olive" title="b2">b</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>  st <font color="Olive" title="b2">b</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> holds <br/> ex <font color="Olive" title="b3">q</font>, <font color="Olive" title="b4">r</font> being  <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> st <br/>( <font color="Olive" title="b1">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b3">q</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b2">b</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b4">r</font> &amp; ( <font color="Olive" title="b4">r</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> or <font color="Maroon" title="c6">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b4">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a> <font color="Maroon" title="c6">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b2">b</font> ) )
 <b>by </b><i><a class="ref" href="int_3.html#D9" title="INT_3:def.9">INT_3:def 9</a></i>;<br/><b>set </b><font color="Maroon" title="c7">K</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">I'</font> : verum </span>}</span> ;<br/><b>consider </b><font color="Maroon" title="c8">i</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">I'</font>;<br/><a NAME="E10:161_1_2"/><i><font color="Green" title="E68">A6</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c8">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">I'</font> : verum </span>}</span> 
 ;<br/><a NAME="E11:161_1_2"/>
<span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">I'</font> : verum </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="161_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon" title="c9">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><br/>

<b>assume </b><a NAME="E1:161_1_2_2"/>
<font color="Maroon" title="c9">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"><font color="Maroon" title="c6">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">I'</font> : verum </span>}</span> 
 ;<br/>

<b>then consider </b><font color="Maroon" title="c10">e</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">I'</font><b> such that </b><br/><a NAME="E3:161_1_2_2"/><i><font color="Green" title="E69">A7</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c10">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">x</font>
 ;<br/>
<b>thus </b><a NAME="E4:161_1_2_2"/>
<font color="Maroon" title="c9">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="ideal_1.html#E3:161_1_2_2"><i><font color="Green" title="E69">A7</font></i></a></i>;<br/>


</div><b>end;</b></div><b>then reconsider </b><font color="Maroon" title="c9">K</font> = <span class="p1">{<span class="default"> <span class="p2">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">I'</font> : verum </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 <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a>  <b>by </b><i><a class="txt" href="ideal_1.html#E10:161_1_2"><i><font color="Green" title="E68">A6</font></i></a></i>;<br/><b>set </b><font color="Maroon" title="c10">k</font> =  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c9">K</font>;<br/><a NAME="E13:161_1_2"/>
 <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c9">K</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">K</font>
 <b>by </b><i><a class="ref" href="seq_4.html#D6" title="SEQ_4:def.6">SEQ_4:def 6</a></i>;<br/><b>then consider </b><font color="Maroon" title="c11">e</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c5">I'</font><b> such that </b><br/><a NAME="E15:161_1_2"/><i><font color="Green" title="E69">A8</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c11">e</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c9">K</font>
 ;<br/><a NAME="E16:161_1_2"/>
<font color="Maroon" title="c11">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I'</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c12">e'</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">I</font><b> such that </b><br/><a NAME="E18:161_1_2"/><i><font color="Green" title="E70">A9</font></i>: 
( <font color="Maroon" title="c12">e'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c11">e</font> &amp; <font color="Maroon" title="c12">e'</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font> )
 ;<br/><b>reconsider </b><font color="Maroon" title="c13">e</font> = <font color="Maroon" title="c11">e</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/><b>take </b>
<font color="Maroon" title="c13">e</font>
; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D28" target="_self" title="IDEAL_1:def.28">IDEAL_1:def 28</a><br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="161_1_2_3"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c14">x</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><div><b>hereby </b>
<div class="add"><b>assume </b><a NAME="E1:161_1_2_3_1"/><i><font color="Green" title="E71">A10</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ;<br/><b>then reconsider </b><font color="Maroon" title="c15">x'</font> = <font color="Maroon" title="c14">x</font> as   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> ;<br/><b>consider </b><font color="Maroon" title="c16">q</font>, <font color="Maroon" title="c17">r</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font><b> such that </b><br/><a NAME="E4:161_1_2_3_1"/><i><font color="Green" title="E72">A11</font></i>: 
<font color="Maroon" title="c15">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">q</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">e</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c17">r</font>
 <b>and </b><br/><a NAME="E5:161_1_2_3_1"/><i><font color="Green" title="E73">A12</font></i>: 
( <font color="Maroon" title="c17">r</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> or <font color="Maroon" title="c6">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c17">r</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="seq_4.html#K6" title="SEQ_4:func.6">min</a> <font color="Maroon" title="c9">K</font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E8:161_1_2"><i><font color="Green" title="E67">A5</font></i></a>, <a class="txt" href="ideal_1.html#E15:161_1_2"><i><font color="Green" title="E69">A8</font></i></a>, <a class="txt" href="ideal_1.html#E18:161_1_2"><i><font color="Green" title="E70">A9</font></i></a></i>;<br/><div><a class="txt" onclick="hs2(this)" href="javascript:()" title="161_1_2_3_1_1"><b>now </b></a><div class="add"><i><font color="Green" title="E74">A13</font></i>: <span class="p1">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">q</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">e</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c15">x'</font> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p3">(<span class="default"><font color="Maroon" title="c16">q</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">e</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p2">(<span class="default"><font color="Maroon" title="c16">q</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">e</font></span>)</span></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c17">r</font>
<b>by </b><i><a class="txt" href="ideal_1.html#E4:161_1_2_3_1"><i><font color="Green" title="E72">A11</font></i></a>, <a class="ref" href="rlvect_1.html#D6" title="RLVECT_1:def.6">RLVECT_1:def 6</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="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Maroon" title="c17">r</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T16" title="RLVECT_1:th.16">RLVECT_1:16</a></i>
<br/>.= 
<font color="Maroon" title="c17">r</font>
<b>by </b><i><a class="ref" href="algstr_1.html#D5" title="ALGSTR_1:def.5">ALGSTR_1:def 5</a></i>
;<br/><a NAME="E2:161_1_2_3_1_1"/>
<font color="Maroon" title="c16">q</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E18:161_1_2"><i><font color="Green" title="E70">A9</font></i></a>, <a class="ref" href="ideal_1.html#D2" target="_self" title="IDEAL_1:def.2">Def2</a></i>;<br/><a NAME="E3:161_1_2_3_1_1"/><b>then </b>
 <a href="rlvect_1.html#K4" title="RLVECT_1:func.4">-</a> <span class="p1">(<span class="default"><font color="Maroon" title="c16">q</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">e</font></span>)</span> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#T13" target="_self" title="IDEAL_1:th.13">Th13</a></i>;<br/><a NAME="E4:161_1_2_3_1_1"/><b>then </b><i><font color="Green" title="E75">A14</font></i>: 
<font color="Maroon" title="c17">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:161_1_2_3_1"><i><font color="Green" title="E71">A10</font></i></a>, <a class="txt" href="ideal_1.html#E1:161_1_2_3_1_1"><i><font color="Green" title="E74">A13</font></i></a>, <a class="ref" href="ideal_1.html#D1" target="_self" title="IDEAL_1:def.1">Def1</a></i>;<br/><b>assume </b><a NAME="E5:161_1_2_3_1_1"/><i><font color="Green" title="E76">A15</font></i>: 
<font color="Maroon" title="c17">r</font> <a href="hidden.html#NR2" title="HIDDEN:NR.2">&lt;&gt;</a>  <a href="struct_0.html#K4" title="STRUCT_0:func.4">0.</a> <font color="Maroon" title="c1">R</font>
 ;<br/><a NAME="E6:161_1_2_3_1_1"/><b>then </b>
<font color="Maroon" title="c17">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c5">I'</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:161_1_2_3_1_1"><i><font color="Green" title="E75">A14</font></i></a></i>;<br/><a NAME="E7:161_1_2_3_1_1"/><b>then </b>
<font color="Maroon" title="c6">f</font> <a href="polynom1.html#K8" title="POLYNOM1:func.8">.</a> <font color="Maroon" title="c17">r</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c9">K</font>
 ;<br/><b>hence </b><a NAME="E8:161_1_2_3_1_1"/>
contradiction
 <b>by </b><i><a class="txt" href="ideal_1.html#E5:161_1_2_3_1"><i><font color="Green" title="E73">A12</font></i></a>, <a class="txt" href="ideal_1.html#E5:161_1_2_3_1_1"><i><font color="Green" title="E76">A15</font></i></a>, <a class="ref" href="seq_4.html#D6" title="SEQ_4:def.6">SEQ_4:def 6</a></i>;<br/></div><b>end;</b></div><a NAME="E7:161_1_2_3_1"/><b>then </b><i><font color="Green" title="E74">A16</font></i>: 
<font color="Maroon" title="c15">x'</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c16">q</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c13">e</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:161_1_2_3_1"><i><font color="Green" title="E72">A11</font></i></a>, <a class="ref" href="rlvect_1.html#D7" title="RLVECT_1:def.7">RLVECT_1:def 7</a></i>;<br/><a NAME="E8:161_1_2_3_1"/><i><font color="Green" title="E75">A17</font></i>: 
<font color="Maroon" title="c13">e</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span>
 <b>by </b><i><a class="ref" href="tarski.html#D1" title="TARSKI:def.1">TARSKI:def 1</a></i>;<br/><a NAME="E9:161_1_2_3_1"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <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" target="_self" title="IDEAL_1:def.15">Def15</a></i>;<br/><b>hence </b><a NAME="E10:161_1_2_3_1"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:161_1_2_3_1"><i><font color="Green" title="E74">A16</font></i></a>, <a class="txt" href="ideal_1.html#E8:161_1_2_3_1"><i><font color="Green" title="E75">A17</font></i></a>, <a class="ref" href="ideal_1.html#D2" target="_self" title="IDEAL_1:def.2">Def2</a></i>;<br/></div>
<b>end;</b></div><b>assume </b><a NAME="E2:161_1_2_3"/><i><font color="Green" title="E71">A18</font></i>: 
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 ;<br/><a NAME="E3:161_1_2_3"/>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E18:161_1_2"><i><font color="Green" title="E70">A9</font></i></a>, <a class="ref" href="zfmisc_1.html#T37" title="ZFMISC_1:th.37">ZFMISC_1:37</a></i>;<br/><a NAME="E4:161_1_2_3"/><b>then </b>
<span class="p1"><a href="domain_1.html#K6" title="DOMAIN_1:func.6">{</a><span class="default"><font color="Maroon" title="c13">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a>  <a href="tarski.html#R1" title="TARSKI:pred.1">c=</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D15" target="_self" title="IDEAL_1:def.15">Def15</a></i>;<br/><b>hence </b><a NAME="E5:161_1_2_3"/>
<font color="Maroon" title="c14">x</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E2:161_1_2_3"><i><font color="Green" title="E71">A18</font></i></a></i>;<br/></div><b>end;</b></div><b>hence </b><a NAME="E21:161_1_2"/>
<font color="Maroon" title="c2">I</font> <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"><font color="Maroon" title="c13">e</font></span><a href="domain_1.html#K6" title="DOMAIN_1:func.6">}</a></span> <a href="ideal_1.html#K7" title="IDEAL_1:func.7">-Ideal</a> 
 <b>by </b><i><a class="ref" href="tarski.html#T2" title="TARSKI:th.2">TARSKI:2</a></i>;<br/></div><b>end;</b></div></div><b>end;</b></div>

</div>
