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<div class="add">

<b>let </b><font color="Maroon">R</font> be   <a href="funcsdom.html#NM1">Ring</a>;<br/><b>let </b><font color="Maroon">I</font> be   <a href="ideal_1.html#NM1">Ideal</a> of <font color="Maroon">R</font>;<br/>



<b>defpred </b><font color="Maroon">S<sub>1</sub></font>[   <a href="hidden.html#M1">set</a> ,   <a href="hidden.html#M1">set</a> ] <b>means </b> ex <font color="Olive">P</font>, <font color="Olive">Q</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>1</sub></font> &amp; <font color="Olive">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">a<sub>2</sub></font> &amp; <font color="Olive">P</font> <a href="rlvect_1.html#K6">-</a> <font color="Olive">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> );<br/>
<a NAME="E1:14"/><i><font color="Green">E42</font></i>: 
for <font color="Olive">x</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:14_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font>
 ;<br/>

<b>then reconsider </b><font color="Maroon">x</font> = <font color="Maroon">x</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> ;<br/>
<a NAME="E3:14_1"/><i><font color="Green">E47</font></i>: 
<font color="Maroon">x</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T28">RLVECT_1:28</a></i>;<br/>
<a NAME="E4:14_1"/>
 <a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font>
 
<b>by </b><i><a class="ref" href="ideal_1.html#T2">IDEAL_1:2</a></i>;<br/>
<b>hence </b><a NAME="E5:14_1"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">x</font>]
 <b>by </b><i/>;<br/>


</div><b>end;</b></div>
<a NAME="E2:14"/><i><font color="Green">E48</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">x</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_2"><b>proof </b></a><div class="add">

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

<b>given </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">Q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><a NAME="E1:14_2"/><i><font color="Green">E49</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">P</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 ;<br/>

<b>take </b>
<font color="Maroon">Q</font>
;<br/>

<b>take </b>
<font color="Maroon">P</font>
;<br/>

<a NAME="E2:14_2"/>
 <a href="rlvect_1.html#K5">-</a> <span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">Q</font></span>)</span> <a href="hidden.html#R1">=</a> <font color="Maroon">Q</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">P</font>
 
<b>by </b><i><a class="ref" href="rlvect_1.html#T47">RLVECT_1:47</a></i>;<br/>
<b>hence </b><a NAME="E3:14_2"/>
( <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">Q</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">P</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 <b>by </b><i>, <a class="ref" href="ideal_1.html#T13">IDEAL_1:13</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E3:14"/><i><font color="Green">E50</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font>, <font color="Olive">z</font> being   <a href="hidden.html#M1">set</a>   st <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">y</font>,<font color="Olive">z</font>] holds <br/><font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">z</font>]
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="14_3"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:14_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">x</font>,<font color="Maroon">y</font>]
 ;<br/>

<b>then consider </b><font color="Maroon">P</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">Q</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E3:14_3"/><i><font color="Green">E52</font></i>: 
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">Q</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">P</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">Q</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 ;<br/>
<b>assume </b><a NAME="E4:14_3"/>
<font color="Maroon">S<sub>1</sub></font>[<font color="Maroon">y</font>,<font color="Maroon">z</font>]
 ;<br/>

<b>then consider </b><font color="Maroon">W</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">S</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E6:14_3"/><i><font color="Green">E54</font></i>: 
( <font color="Maroon">W</font> <a href="hidden.html#R1">=</a> <font color="Maroon">y</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">W</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 ;<br/>
<b>take </b>
<font color="Maroon">P</font>
;<br/>

<b>take </b>
<font color="Maroon">S</font>
;<br/>

<span class="p1">(<span class="default"><font color="Maroon">P</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">Q</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">Q</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">S</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">P</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">Q</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">Q</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">S</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T42">RLVECT_1:42</a></i>
<br/>.= 
<font color="Maroon">P</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">S</font>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E8:14_3"/>
( <font color="Maroon">P</font> <a href="hidden.html#R1">=</a> <font color="Maroon">x</font> &amp; <font color="Maroon">S</font> <a href="hidden.html#R1">=</a> <font color="Maroon">z</font> &amp; <font color="Maroon">P</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">S</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 <b>by </b><i>, , <a class="ref" href="ideal_1.html#D1">IDEAL_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>thus </b><a NAME="E4:14"/>
 ex <font color="Olive">EqR</font> being  <a href="eqrel_1.html#NM2">Equivalence_Relation</a> of the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> st <br/>for <font color="Olive">x</font>, <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>  holds <br/> ( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">x</font>,<font color="Olive">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Olive">EqR</font> iff ( <font color="Olive">x</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">R</font> &amp; <font color="Maroon">S<sub>1</sub></font>[<font color="Olive">x</font>,<font color="Olive">y</font>] ) )
 <b>from </b><i><a class="ref" href="eqrel_1.html#S1">EQREL_1:sch 1</a>(, , );</i><br/>


</div>
