<?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="vectsp_1.html#V4" title="VECTSP_1:attr.4">right-distributive</a> <a href="algstr_1.html#V1" title="ALGSTR_1:attr.1">left_zeroed</a> <a href="algstr_1.html#V3" title="ALGSTR_1:attr.3">add-right-cancelable</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  non <a href="xboole_0.html#V1" title="XBOOLE_0:attr.1">empty</a> <a href="ideal_1.html#V1" title="IDEAL_1:attr.1">add-closed</a> <a href="ideal_1.html#V3" title="IDEAL_1:attr.3">right-ideal</a> <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>;<br/><b>let </b><font color="Maroon" title="c3">J</font> be   <a href="struct_0.html#NM2" title="STRUCT_0:NM.2">Subset</a> of <font color="Maroon" title="c1">R</font>;<br/>





<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="143_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c4">u</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/><b>assume </b><a NAME="E1:143_1"/>
<font color="Maroon" title="c4">u</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K16" title="IDEAL_1:func.16">%</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="ideal_1.html#K14" title="IDEAL_1:func.14">*'</a> <font color="Maroon" title="c3">J</font>
 ;<br/><b>then consider </b><font color="Maroon" title="c5">s</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="E3:143_1"/><i><font color="Green" title="E59">A1</font></i>: 
(  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c4">u</font> &amp; ( for <font color="Olive" title="b1">i</font> being   <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>   st 1 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">i</font> &amp; <font color="Olive" title="b1">i</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s</font> holds <br/> ex <font color="Olive" title="b2">a</font>, <font color="Olive" title="b3">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 <br/>( <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Olive" title="b1">i</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Olive" title="b2">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b3">b</font> &amp; <font color="Olive" title="b2">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K16" title="IDEAL_1:func.16">%</a> <font color="Maroon" title="c3">J</font> &amp; <font color="Olive" title="b3">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> ) ) )
 ;<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  <a href="numbers.html#K5" title="NUMBERS:func.5">NAT</a> ,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="E5:143_1"/><i><font color="Green" title="E60">A2</font></i>: 
(  <a href="rlvect_1.html#K8" title="RLVECT_1:func.8">Sum</a> <font color="Maroon" title="c5">s</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s</font></span>)</span> &amp; <font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> 0 <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> &amp; ( for <font color="Olive" title="b1">j</font> being   <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/> for <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>  st <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s</font> &amp; <font color="Olive" title="b2">v</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> holds <br/><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Olive" title="b1">j</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <font color="Olive" title="b2">v</font> ) )
 <b>by </b><i><a class="ref" href="rlvect_1.html#D12" title="RLVECT_1:def.12">RLVECT_1:def 12</a></i>;<br/><b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <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>means </b><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> $1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>;<br/><a NAME="E6:143_1"/><i><font color="Green" title="E61">A3</font></i>: 
<font color="Maroon">S<sub>1</sub></font>[0]
 <b>by </b><i><a class="txt" href="ideal_1.html#E5:143_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="ref" href="ideal_1.html#T3" target="_self" title="IDEAL_1:th.3">Th3</a></i>;<br/><div><i><font color="Green" title="E62">A4</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="143_1_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c7">j</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> ;<br/><b>assume </b><a NAME="E1:143_1_1"/><i><font color="Green" title="E63">A5</font></i>: 
( 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Maroon" title="c7">j</font> &amp; <font color="Maroon" title="c7">j</font> <a href="xxreal_0.html#NR3" title="XXREAL_0:NR.3">&lt;</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s</font> )
 ;<br/><b>thus </b><a NAME="E2:143_1_1"/>
( <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">j</font>] implies <font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1] )
 <div><a class="txt" onclick="hs2(this)" href="javascript:()" title="143_1_1_1"><b>proof </b></a><div class="add">

<b>assume </b><a NAME="E1:143_1_1_1"/><i><font color="Green" title="E64">A6</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">j</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font>
 ;<br/>

<a NAME="E2:143_1_1_1"/><i><font color="Green" title="E65">A7</font></i>: 
<font color="Maroon" title="c7">j</font> <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="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E1:143_1_1"><i><font color="Green" title="E63">A5</font></i></a>, <a class="ref" href="nat_1.html#T13" title="NAT_1:th.13">NAT_1:13</a></i>;<br/>
<a NAME="E3:143_1_1_1"/><i><font color="Green" title="E66">A8</font></i>: 
0 <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> <font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1
 
<b>by </b><i><a class="ref" href="xreal_1.html#T8" title="XREAL_1:th.8">XREAL_1:8</a></i>;<br/>
<a NAME="E4:143_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <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="c5">s</font></span>)</span>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E2:143_1_1_1"><i><font color="Green" title="E65">A7</font></i></a>, <a class="ref" href="finseq_1.html#T3" title="FINSEQ_1:th.3">FINSEQ_1:3</a></i>;<br/>
<a NAME="E5:143_1_1_1"/><b>then </b>
<font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1 <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a>  <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">s</font>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<a NAME="E6:143_1_1_1"/><b>then </b><i><font color="Green" title="E67">A9</font></i>: 
<font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c5">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_4.html#D4" title="FINSEQ_4:def.4">FINSEQ_4:def 4</a></i>;<br/>
<a NAME="E7:143_1_1_1"/><b>then </b><i><font color="Green" title="E68">A10</font></i>: 
<font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">f</font> <a href="funct_2.html#K3" title="FUNCT_2:func.3">.</a> <font color="Maroon" title="c7">j</font></span>)</span> <a href="rlvect_1.html#K1" title="RLVECT_1:func.1">+</a> <span class="p1">(<span class="default"><font color="Maroon" title="c5">s</font> <a href="finseq_4.html#K4" title="FINSEQ_4:func.4">/.</a> <span class="p2">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span></span>)</span>
 
<b>by </b><i><a class="txt" href="ideal_1.html#E5:143_1"><i><font color="Green" title="E60">A2</font></i></a>, <a class="txt" href="ideal_1.html#E1:143_1_1"><i><font color="Green" title="E63">A5</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">a</font>, <font color="Maroon" title="c9">b</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="E9:143_1_1_1"/><i><font color="Green" title="E69">A11</font></i>: 
( <font color="Maroon" title="c5">s</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <span class="p1">(<span class="default"><font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1</span>)</span> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">b</font> &amp; <font color="Maroon" title="c8">a</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K16" title="IDEAL_1:func.16">%</a> <font color="Maroon" title="c3">J</font> &amp; <font color="Maroon" title="c9">b</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:143_1"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="ideal_1.html#E2:143_1_1_1"><i><font color="Green" title="E65">A7</font></i></a>, <a class="txt" href="ideal_1.html#E3:143_1_1_1"><i><font color="Green" title="E66">A8</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c10">d</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="E11:143_1_1_1"/><i><font color="Green" title="E70">A12</font></i>: 
( <font color="Maroon" title="c8">a</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">d</font> &amp; <font color="Maroon" title="c10">d</font> <a href="ideal_1.html#K10" title="IDEAL_1:func.10">*</a> <font color="Maroon" title="c3">J</font> <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#E9:143_1_1_1"><i><font color="Green" title="E69">A11</font></i></a></i>;<br/>
<a NAME="E12:143_1_1_1"/>
<font color="Maroon" title="c8">a</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Maroon" title="c9">b</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="c10">d</font> <a href="group_1.html#K1" title="GROUP_1:func.1">*</a> <font color="Olive" title="b1">i</font></span>)</span> where <font color="Olive" title="b1">i</font> is   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font> : <font color="Olive" title="b1">i</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c3">J</font> </span>}</span> 
 
<b>by </b><i><a class="txt" href="ideal_1.html#E9:143_1_1_1"><i><font color="Green" title="E69">A11</font></i></a>, <a class="txt" href="ideal_1.html#E11:143_1_1_1"><i><font color="Green" title="E70">A12</font></i></a></i>;<br/>
<b>hence </b><a NAME="E13:143_1_1_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c7">j</font> <a href="nat_1.html#K1" title="NAT_1:func.1">+</a> 1]
 <b>by </b><i><a class="txt" href="ideal_1.html#E1:143_1_1_1"><i><font color="Green" title="E64">A6</font></i></a>, <a class="txt" href="ideal_1.html#E6:143_1_1_1"><i><font color="Green" title="E67">A9</font></i></a>, <a class="txt" href="ideal_1.html#E7:143_1_1_1"><i><font color="Green" title="E68">A10</font></i></a>, <a class="txt" href="ideal_1.html#E9:143_1_1_1"><i><font color="Green" title="E69">A11</font></i></a>, <a class="txt" href="ideal_1.html#E11:143_1_1_1"><i><font color="Green" title="E70">A12</font></i></a>, <a class="ref" href="ideal_1.html#D1" target="_self" title="IDEAL_1:def.1">Def1</a></i>;<br/>


</div><b>end;</b></div></div><b>end;</b></div><a NAME="E8:143_1"/>
for <font color="Olive" title="b1">j</font> being   <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>   st 0 <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a> <font color="Olive" title="b1">j</font> &amp; <font color="Olive" title="b1">j</font> <a href="xxreal_0.html#R1" title="XXREAL_0:pred.1">&lt;=</a>  <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">s</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">j</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S4" title="POLYNOM2:sch.4">POLYNOM2:sch 4</a>(<a class="txt" href="ideal_1.html#E6:143_1"><i><font color="Green" title="E61">A3</font></i></a>, <a class="txt" href="ideal_1.html#E7:143_1"><i><font color="Green" title="E62">A4</font></i></a>);<br/></i><b>hence </b><a NAME="E9:143_1"/>
<font color="Maroon" title="c4">u</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#E3:143_1"><i><font color="Green" title="E59">A1</font></i></a>, <a class="txt" href="ideal_1.html#E5:143_1"><i><font color="Green" title="E60">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>hence </b><a NAME="E2:143"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c2">I</font> <a href="ideal_1.html#K16" title="IDEAL_1:func.16">%</a> <font color="Maroon" title="c3">J</font></span>)</span> <a href="ideal_1.html#K14" title="IDEAL_1:func.14">*'</a> <font color="Maroon" title="c3">J</font> <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="tarski.html#D3" title="TARSKI:def.3">TARSKI:def 3</a></i>;<br/>


</div>
