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<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="algstr_0.html#L4" title="ALGSTR_0:struct.4">multLoopStr</a> ;<br/><b>let </b><font color="Maroon" title="c2">A</font> be  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>;<br/><b>let </b><font color="Maroon" title="c3">L</font> be    <a href="ideal_1.html#M1" title="IDEAL_1:mode.1">LinearCombination</a> of <font color="Maroon" title="c2">A</font>;<br/>





<b>set </b><font color="Maroon" title="c4">D</font> = <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 <font color="Maroon" title="c1">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span>;<br/>
<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ,   <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ] <b>means </b> ex <font color="Olive" title="b1">x</font>, <font color="Olive" title="b2">y</font>, <font color="Olive" title="b3">z</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/>( $2 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Olive" title="b1">x</font>,<font color="Olive" title="b2">y</font>,<font color="Olive" title="b3">z</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Olive" title="b2">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> $1 <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Olive" title="b1">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b2">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Olive" title="b3">z</font> );<br/>
<div><i><font color="Green" title="E33">A1</font></i>: <a class="txt" onclick="hs2(this)" href="javascript:()" title="64_1"><b>now </b></a><div class="add"><b>let </b><font color="Maroon" title="c5">k</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="E1:64_1"/>
<font color="Maroon" title="c5">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="c3">L</font></span>)</span>
 ;<br/><a NAME="E2:64_1"/><b>then </b>
<font color="Maroon" title="c5">k</font> <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="c3">L</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/><b>then consider </b><font color="Maroon" title="c6">u</font>, <font color="Maroon" title="c7">v</font> being   <a href="struct_0.html#NM1" title="STRUCT_0:NM.1">Element</a> of <font color="Maroon" title="c1">R</font>, <font color="Maroon" title="c8">a</font> being    <a href="subset_1.html#M2" title="SUBSET_1:mode.2">Element</a> of <font color="Maroon" title="c2">A</font><b> such that </b><br/><a NAME="E4:64_1"/><i><font color="Green" title="E34">A2</font></i>: 
<font color="Maroon" title="c3">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c5">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c6">u</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c8">a</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c7">v</font>
 <b>by </b><i><a class="ref" href="ideal_1.html#D9" target="_self" title="IDEAL_1:def.9">Def9</a></i>;<br/><b>reconsider </b><font color="Maroon" title="c9">b</font> = <font color="Maroon" title="c8">a</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>reconsider </b><font color="Maroon" title="c10">d</font> = <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c6">u</font>,<font color="Maroon" title="c9">b</font>,<font color="Maroon" title="c7">v</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> as    <a href="subset_1.html#M1" title="SUBSET_1:mode.1">Element</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 <font color="Maroon" title="c1">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span> ;<br/><b>take </b><font color="Maroon" title="c11">d</font> = <font color="Maroon" title="c10">d</font>;<br/><b>thus </b><a NAME="E7:64_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon" title="c5">k</font>,<font color="Maroon" title="c11">d</font>]
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:64_1"><i><font color="Green" title="E34">A2</font></i></a></i>;<br/></div><b>end;</b></div>
<b>consider </b><font color="Maroon" title="c5">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 <font color="Maroon" title="c1">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font>,the <a href="struct_0.html#U1" title="STRUCT_0:sel.1">carrier</a> of <font color="Maroon" title="c1">R</font></span><a href="zfmisc_1.html#K3" title="ZFMISC_1:func.3">:]</a></span><b> such that </b><br/><a NAME="E3:64"/><i><font color="Green" title="E34">A3</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c5">E</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="c3">L</font></span>)</span>
 <b>and </b><br/><a NAME="E4:64"/><i><font color="Green" title="E35">A4</font></i>: 
for <font color="Olive" title="b1">k</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 <font color="Olive" title="b1">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="c3">L</font></span>)</span> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive" title="b1">k</font>,<font color="Maroon" title="c5">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Olive" title="b1">k</font>]
 <b>from </b><i><a class="ref" href="polynom2.html#S1" title="POLYNOM2:sch.1">POLYNOM2:sch 1</a>(<a class="txt" href="ideal_1.html#E1:64"><i><font color="Green" title="E33">A1</font></i></a>);<br/></i>
<b>take </b>
<font color="Maroon" title="c5">E</font>
;<br/>

<a NAME="E5:64"/><i><font color="Green" title="E36">A5</font></i>: 
 <a href="relset_1.html#K4" title="RELSET_1:func.4">dom</a> <font color="Maroon" title="c3">L</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="c3">L</font></span>)</span>
 
<b>by </b><i><a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>;<br/>
<b>thus </b><a NAME="E6:64"/>
 <a href="finseq_1.html#K3" title="FINSEQ_1:func.3">len</a> <font color="Maroon" title="c5">E</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="c3">L</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E3:64"><i><font color="Green" title="E34">A3</font></i></a>, <a class="ref" href="finseq_1.html#D3" title="FINSEQ_1:def.3">FINSEQ_1:def 3</a></i>; <i><font color="Red">:: according to </font></i><a class="ref" href="ideal_1.html#D12" target="_self" title="IDEAL_1:def.12">IDEAL_1:def 12</a><br/>

<b>let </b><font color="Maroon" title="c6">i</font> be    <a href="hidden.html#M1" title="HIDDEN:mode.1">set</a> ;<br/>

<b>assume </b><a NAME="E7:64"/><i><font color="Green" title="E37">A6</font></i>: 
<font color="Maroon" title="c6">i</font> <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="c3">L</font>
 ;<br/>

<b>reconsider </b><font color="Maroon" title="c7">k</font> = <font color="Maroon" title="c6">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>  <b>by </b><i><a class="txt" href="ideal_1.html#E7:64"><i><font color="Green" title="E37">A6</font></i></a></i>;<br/>
<b>consider </b><font color="Maroon" title="c8">x</font>, <font color="Maroon" title="c9">y</font>, <font color="Maroon" title="c10">z</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="E10:64"/><i><font color="Green" title="E38">A7</font></i>: 
( <font color="Maroon" title="c5">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">z</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> &amp; <font color="Maroon" title="c9">y</font> <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font> &amp; <font color="Maroon" title="c3">L</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c7">k</font> <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <span class="p1">(<span class="default"><font color="Maroon" title="c8">x</font> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c9">y</font></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <font color="Maroon" title="c10">z</font> )
 <b>by </b><i><a class="txt" href="ideal_1.html#E4:64"><i><font color="Green" title="E35">A4</font></i></a>, <a class="txt" href="ideal_1.html#E5:64"><i><font color="Green" title="E36">A5</font></i></a>, <a class="txt" href="ideal_1.html#E7:64"><i><font color="Green" title="E37">A6</font></i></a></i>;<br/>
<a NAME="E11:64"/>
( <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">z</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c8">x</font> &amp; <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">z</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c9">y</font> &amp; <span class="p1"><a href="domain_1.html#K4" title="DOMAIN_1:func.4">[</a><span class="default"><font color="Maroon" title="c8">x</font>,<font color="Maroon" title="c9">y</font>,<font color="Maroon" title="c10">z</font></span><a href="domain_1.html#K4" title="DOMAIN_1:func.4">]</a></span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a>  <a href="hidden.html#R1" title="HIDDEN:pred.1">=</a> <font color="Maroon" title="c10">z</font> )
 
<b>by </b><i><a class="ref" href="mcart_1.html#D5" title="MCART_1:def.5">MCART_1:def 5</a>, <a class="ref" href="mcart_1.html#D6" title="MCART_1:def.6">MCART_1:def 6</a>, <a class="ref" href="mcart_1.html#D7" title="MCART_1:def.7">MCART_1:def 7</a></i>;<br/>
<b>hence </b><a NAME="E12:64"/>
<font color="Maroon" title="c3">L</font> <a href="funct_1.html#K1" title="FUNCT_1:func.1">.</a> <font color="Maroon" title="c6">i</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"><font color="Maroon" title="c5">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="mcart_1.html#K5" title="MCART_1:func.5">`1</a> </span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a> </span>)</span></span>)</span> <a href="algstr_0.html#K6" title="ALGSTR_0:func.6">*</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="mcart_1.html#K7" title="MCART_1:func.7">`3</a> </span>)</span>
 <b>by </b><i><a class="txt" href="ideal_1.html#E7:64"><i><font color="Green" title="E37">A6</font></i></a>, <a class="txt" href="ideal_1.html#E10:64"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="partfun1.html#D8" title="PARTFUN1:def.8">PARTFUN1:def 8</a></i>;<br/>

<b>thus </b><a NAME="E13:64"/>
<span class="p1">(<span class="default"><font color="Maroon" title="c5">E</font> <a href="partfun1.html#K7" title="PARTFUN1:func.7">/.</a> <font color="Maroon" title="c6">i</font></span>)</span> <a href="mcart_1.html#K6" title="MCART_1:func.6">`2</a>  <a href="hidden.html#R2" title="HIDDEN:pred.2">in</a> <font color="Maroon" title="c2">A</font>
 <b>by </b><i><a class="txt" href="ideal_1.html#E10:64"><i><font color="Green" title="E38">A7</font></i></a>, <a class="ref" href="mcart_1.html#D6" title="MCART_1:def.6">MCART_1:def 6</a></i>;<br/>


</div>
