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

<b>set </b><font color="Maroon">E</font> =  <a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font>;<br/>
<b>let </b><font color="Maroon">X</font> be  <a href="vectsp_1.html#V3">strict</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> , <font color="Maroon">Y</font> be  <a href="vectsp_1.html#V3">strict</a>  <a href="vectsp_1.html#L3">doubleLoopStr</a> ;<br/>

<b>assume </b><b>that </b><br/><a NAME="E1:23_1_2"/><i><font color="Green">E77</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>
 <b>and </b><br/><a NAME="E2:23_1_2"/><i><font color="Green">E78</font></i>: 
the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>
 <b>and </b><br/><a NAME="E3:23_1_2"/><i><font color="Green">E79</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
 <b>and </b><br/><a NAME="E4:23_1_2"/><i><font color="Green">E80</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Olive">a</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Olive">b</font> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b</font></span>)</span> )
 <b>and </b><br/><a NAME="E5:23_1_2"/><i><font color="Green">E81</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Olive">a</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Olive">b</font> &amp; the <a href="group_1.html#U1">mult</a> of <font color="Maroon">X</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font></span>)</span> )
 <b>and </b><br/><a NAME="E6:23_1_2"/><i><font color="Green">E82</font></i>: 
the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K9">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>
 <b>and </b><br/><a NAME="E7:23_1_2"/><i><font color="Green">E83</font></i>: 
the <a href="vectsp_1.html#U1">unity</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">R</font></span>)</span>
 <b>and </b><br/><a NAME="E8:23_1_2"/><i><font color="Green">E84</font></i>: 
the <a href="struct_0.html#U2">Zero</a> of <font color="Maroon">Y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><a href="rlvect_1.html#K1">0.</a> <font color="Maroon">R</font></span>)</span>
 <b>and </b><br/><a NAME="E9:23_1_2"/><i><font color="Green">E85</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Olive">a</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Olive">b</font> &amp; the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Olive">b</font></span>)</span> )
 <b>and </b><br/><a NAME="E10:23_1_2"/><i><font color="Green">E86</font></i>: 
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">Y</font>  ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font> st <br/>( <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Olive">a</font> &amp; <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Olive">b</font> &amp; the <a href="group_1.html#U1">mult</a> of <font color="Maroon">Y</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Olive">a</font> <a href="group_1.html#K1">*</a> <font color="Olive">b</font></span>)</span> )
 ;<br/>

<a NAME="E11:23_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E2:23_1_2_1"/><i><font color="Green">E87</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">b</font> )
 <b>and </b><br/><a NAME="E3:23_1_2_1"/><i><font color="Green">E88</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:23_1_2_1"/><i><font color="Green">E91</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">a1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">b1</font> )
 <b>and </b><br/><a NAME="E6:23_1_2_1"/><i><font color="Green">E92</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b1</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>
<a NAME="E7:23_1_2_1"/>
( <font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">b</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 
<b>by </b><i>, , </i>;<br/>
<a NAME="E8:23_1_2_1"/><b>then </b><i><font color="Green">E93</font></i>: 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b1</font></span>)</span> <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/>
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b1</font></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a1</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T41">RLVECT_1:41</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b1</font>
<b>by </b><i><a class="ref" href="rlvect_1.html#T42">RLVECT_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a1</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T42">RLVECT_1:42</a></i>
;<br/>
<b>hence </b><a NAME="E10:23_1_2_1"/>
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i><a class="ref" href="ring_1.html#T1" target="_self">Th1</a>, , , </i>;<br/>


</div><b>end;</b></div>
<a NAME="E12:23_1_2"/><b>then </b><i><font color="Green">E94</font></i>: 
the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">X</font> <a href="hidden.html#R1">=</a> the <a href="rlvect_1.html#U1">add</a> of <font color="Maroon">Y</font>
 
<b>by </b><i>, , <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>
<a NAME="E13:23_1_2"/>
for <font color="Olive">x</font>, <font color="Olive">y</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font> holds  the <a href="group_1.html#U1">mult</a> of <font color="Maroon">X</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">Y</font> <a href="binop_1.html#K1">.</a> <font color="Olive">x</font>,<font color="Olive">y</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="23_1_2_2"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">x</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>, <font color="Maroon">y</font> be   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">X</font>;<br/>

<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E2:23_1_2_2"/><i><font color="Green">E95</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">a</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">b</font> )
 <b>and </b><br/><a NAME="E3:23_1_2_2"/><i><font color="Green">E96</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">X</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span>
 <b>by </b><i/>;<br/>
<b>consider </b><font color="Maroon">a1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font>, <font color="Maroon">b1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">R</font><b> such that </b><br/><a NAME="E5:23_1_2_2"/><i><font color="Green">E97</font></i>: 
( <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">a1</font> &amp; <font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<font color="Maroon">b1</font> )
 <b>and </b><br/><a NAME="E6:23_1_2_2"/><i><font color="Green">E98</font></i>: 
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">Y</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a>  <a href="eqrel_1.html#K7">Class</a> <span class="p1">(<span class="default"><a href="ring_1.html#K1" target="_self">EqRel</a> <font color="Maroon">R</font>,<font color="Maroon">I</font></span>)</span>,<span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b1</font></span>)</span>
 <b>by </b><i>, , </i>;<br/>
<a NAME="E7:23_1_2_2"/>
( <font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">b</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b1</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="ring_1.html#T1" target="_self">Th1</a>, , </i>;<br/>
<a NAME="E8:23_1_2_2"/><b>then </b>
( <span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a1</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> &amp; <font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <span class="p1">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b1</font></span>)</span> <a href="hidden.html#R2">in</a> <font color="Maroon">I</font> )
 
<b>by </b><i><a class="ref" href="ideal_1.html#D2">IDEAL_1:def 2</a>, <a class="ref" href="ideal_1.html#D3">IDEAL_1:def 3</a></i>;<br/>
<a NAME="E9:23_1_2_2"/><b>then </b><i><font color="Green">E99</font></i>: 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a1</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b1</font></span>)</span></span>)</span> <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/>
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">a1</font></span>)</span> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b1</font></span>)</span></span>)</span> = 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <span class="p2">(<span class="default"><font color="Maroon">b</font> <a href="rlvect_1.html#K6">-</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T45">VECTSP_1:45</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p1">(<span class="default"><span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b1</font></span>)</span></span>)</span>
<b>by </b><i><a class="ref" href="vectsp_1.html#T43">VECTSP_1:43</a></i>
<br/>.= 
<span class="p1">(<span class="default"><span class="p2">(<span class="default"><span class="p3">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p3">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K4">+</a> <span class="p2">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i><a class="ref" href="rlvect_1.html#T42">RLVECT_1:42</a></i>
<br/>.= 
<span class="p1">(<span class="default"><font color="Maroon">a</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b</font></span>)</span> <a href="rlvect_1.html#K6">-</a> <span class="p1">(<span class="default"><font color="Maroon">a1</font> <a href="group_1.html#K1">*</a> <font color="Maroon">b1</font></span>)</span>
<b>by </b><i/>
;<br/>
<b>hence </b><a NAME="E11:23_1_2_2"/>
the <a href="group_1.html#U1">mult</a> of <font color="Maroon">X</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font> <a href="hidden.html#R1">=</a> the <a href="group_1.html#U1">mult</a> of <font color="Maroon">Y</font> <a href="binop_1.html#K1">.</a> <font color="Maroon">x</font>,<font color="Maroon">y</font>
 <b>by </b><i>, , <a class="ref" href="ring_1.html#T2" target="_self">Th2</a>, </i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E14:23_1_2"/>
<font color="Maroon">X</font> <a href="hidden.html#R1">=</a> <font color="Maroon">Y</font>
 <b>by </b><i>, , , , , , , <a class="ref" href="binop_1.html#T2">BINOP_1:2</a></i>;<br/>


</div>
