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

<b>set </b><font color="Maroon">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p4">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> </span>}</span> ;<br/>
<a NAME="E1:96_1_1"/><i><font color="Green">E20</font></i>: 
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p4">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> </span>}</span>  is <a href="relat_1.html#V1">Relation-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1"><b>proof </b></a><div class="add">

<a NAME="E1:96_1_1_1"/>
for <font color="Olive">u</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p4">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> </span>}</span>  holds <br/> ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <font color="Olive">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_1_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:96_1_1_1_1"/>
<font color="Maroon">u</font> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p4">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> </span>}</span> 
 ;<br/>

<a NAME="E2:96_1_1_1_1"/><b>then </b>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> st <br/>( <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">a</font>,<span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p3">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">a</font>,<font color="Olive">b</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Olive">b</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> )
 
;<br/>
<b>hence </b><a NAME="E3:96_1_1_1_1"/>
 ex <font color="Olive">a</font>, <font color="Olive">b</font> being   <a href="hidden.html#M1">set</a>  st <font color="Maroon">u</font> <a href="hidden.html#R1">=</a> <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b</font></span><a href="tarski.html#K4">]</a></span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:96_1_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p4">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> </span>}</span>  is <a href="relat_1.html#V1">Relation-like</a>
 <b>by </b><i><a class="ref" href="relat_1.html#D1">RELAT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E2:96_1_1"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p4">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> </span>}</span>  is <a href="funct_1.html#V1">Function-like</a>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2"><b>proof </b></a><div class="add">

<a NAME="E1:96_1_1_2"/>
for <font color="Olive">a</font>, <font color="Olive">b1</font>, <font color="Olive">b2</font> being   <a href="hidden.html#M1">set</a>   st <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p4">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> </span>}</span>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Olive">a</font>,<font color="Olive">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p4">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> </span>}</span>  holds <br/><font color="Olive">b1</font> <a href="hidden.html#R1">=</a> <font color="Olive">b2</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_2_1"><b>proof </b></a><div class="add">

<b>let </b><font color="Maroon">a</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b1</font> be    <a href="hidden.html#M1">set</a> , <font color="Maroon">b2</font> be    <a href="hidden.html#M1">set</a> ;<br/>

<b>assume </b><a NAME="E1:96_1_1_2_1"/><i><font color="Green">E23</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p4">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> </span>}</span>  &amp; <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p4">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> </span>}</span>  )
 ;<br/>

<b>then consider </b><font color="Maroon">x1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">x2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E3:96_1_1_2_1"/><i><font color="Green">E24</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b1</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p3">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">x2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> )
 ;<br/>
<b>consider </b><font color="Maroon">y1</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">y2</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:96_1_1_2_1"/><i><font color="Green">E25</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p3">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">y2</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> )
 <b>by </b><i><a class="ref" href="quofield.html#D9" target="_self">Def9</a></i>;<br/>
<i><font color="Green">E34</font></i>: <font color="Maroon">a</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x1</font>,<span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p3">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Maroon">x1</font>,<font color="Maroon">x2</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="domain_1.html#K2">`1</a> 
<b>by </b><i><a class="ref" href="quofield.html#D10" target="_self">Def10</a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">x1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<i><font color="Green">E39</font></i>: <font color="Maroon">a</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">y1</font>,<span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p3">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Maroon">y1</font>,<font color="Maroon">y2</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="domain_1.html#K2">`1</a> 
<b>by </b><i><a class="ref" href="quofield.html#D11" target="_self">Def11</a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">y1</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>reconsider </b><font color="Maroon">a</font> = <font color="Maroon">a</font> as   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> <b>by </b><i><a class="ref" href="quofield.html#T13" target="_self">Th13</a></i>;<br/>
<font color="Maroon">b1</font> = 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">a</font>,<font color="Maroon">b2</font></span><a href="tarski.html#K4">]</a></span> <a href="mcart_1.html#K2">`2</a> 
<b>by </b><i><a class="ref" href="quofield.html#D10" target="_self">Def10</a>, <a class="ref" href="quofield.html#D11" target="_self">Def11</a>, <a class="ref" href="quofield.html#T13" target="_self">Th13</a>, <a class="ref" href="quofield.html#T14" target="_self">Th14</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
<font color="Maroon">b2</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<b>hence </b><a NAME="E10:96_1_1_2_1"/>
<font color="Maroon">b1</font> <a href="hidden.html#R1">=</a> <font color="Maroon">b2</font>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:96_1_1_2"/>
<span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p4">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> </span>}</span>  is <a href="funct_1.html#V1">Function-like</a>
 <b>by </b><i><a class="ref" href="funct_1.html#D1">FUNCT_1:def 1</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <span class="p1">{<span class="default"> <span class="p2"><a href="domain_1.html#K1">[</a><span class="default"><font color="Olive">x</font>,<span class="p3">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p4">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<font color="Olive">y</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> where <font color="Olive">x</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Olive">y</font> is   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> : <font color="Olive">y</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> </span>}</span>  as   <a href="funct_1.html#NM1">Function</a> <b>by </b><i><a class="ref" href="quofield.html#D8" target="_self">Def8</a></i>;<br/>
<a NAME="E4:96_1_1"/><i><font color="Green">E40</font></i>: 
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_3"><b>proof </b></a><div class="add">

<a NAME="E1:96_1_1_3"/><i><font color="Green">E41</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>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> holds <br/><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">I</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_3_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="E1:96_1_1_3_1"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">y</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:96_1_1_3_1"/><i><font color="Green">E53</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:96_1_1_3_1"/><i><font color="Green">E54</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p3">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> )
 <b>by </b><i><a class="ref" href="quofield.html#D11" target="_self">Def11</a></i>;<br/>
<font color="Maroon">x</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p3">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="domain_1.html#K2">`1</a> 
<b>by </b><i><a class="ref" href="quofield.html#T13" target="_self">Th13</a>, <a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
<br/>.= 
<font color="Maroon">a</font>
<b>by </b><i><a class="ref" href="mcart_1.html#D1">MCART_1:def 1</a></i>
;<br/>
<b>hence </b><a NAME="E7:96_1_1_3_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">I</font>
 ;<br/>


</div><b>end;</b></div>
<a NAME="E2:96_1_1_3"/>
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">I</font> holds <br/><font color="Olive">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_3_2"><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="E1:96_1_1_3_2"/>
<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">I</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">I</font> ;<br/>
<a NAME="E3:96_1_1_3_2"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p3">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 
;<br/>
<b>hence </b><a NAME="E4:96_1_1_3_2"/>
<font color="Maroon">x</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D4">RELAT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E3:96_1_1_3"/>
 <a href="relat_1.html#K1">dom</a> <font color="Maroon">f</font> <a href="hidden.html#R1">=</a> the <a href="struct_0.html#U1">carrier</a> of <font color="Maroon">I</font>
 <b>by </b><i><a class="ref" href="quofield.html#D10" target="_self">Def10</a>, <a class="ref" href="tarski.html#T2">TARSKI:2</a></i>;<br/>


</div><b>end;</b></div>
<a NAME="E5:96_1_1"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_4"><b>proof </b></a><div class="add">

<a NAME="E1:96_1_1_4"/>
for <font color="Olive">y</font> being   <a href="hidden.html#M1">set</a>   st <font color="Olive">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> holds <br/><font color="Olive">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_4_1"><b>proof </b></a><div class="add">

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

<b>assume </b><a NAME="E1:96_1_1_4_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font>
 ;<br/>

<b>then consider </b><font color="Maroon">x</font> being    <a href="hidden.html#M1">set</a> <b> such that </b><br/><a NAME="E3:96_1_1_4_1"/><i><font color="Green">E55</font></i>: 
<span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 <b>by </b><i><a class="ref" href="relat_1.html#D5">RELAT_1:def 5</a></i>;<br/>
<b>consider </b><font color="Maroon">a</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font>, <font color="Maroon">b</font> being   <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font><b> such that </b><br/><a NAME="E5:96_1_1_4_1"/><i><font color="Green">E56</font></i>: 
( <span class="p1"><a href="tarski.html#K4">[</a><span class="default"><font color="Maroon">x</font>,<font color="Maroon">y</font></span><a href="tarski.html#K4">]</a></span> <a href="hidden.html#R1">=</a> <span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p3">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> &amp; <font color="Maroon">b</font> <a href="hidden.html#R1">=</a>  <a href="group_1.html#K2">1.</a> <font color="Maroon">I</font> )
 <b>by </b><i><a class="ref" href="quofield.html#D10" target="_self">Def10</a></i>;<br/>
<font color="Maroon">y</font> = 
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">a</font>,<span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p3">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="domain_1.html#K3">`2</a> 
<b>by </b><i><a class="ref" href="quofield.html#D11" target="_self">Def11</a>, <a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
<br/>.= 
 <a href="quofield.html#K12" target="_self">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Maroon">a</font>,<font color="Maroon">b</font></span>)</span>
<b>by </b><i><a class="ref" href="mcart_1.html#D2">MCART_1:def 2</a></i>
;<br/>
<a NAME="E7:96_1_1_4_1"/><b>then </b>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a>  <a href="quofield.html#K9" target="_self">Quot.</a> <font color="Maroon">I</font>
 
;<br/>
<b>hence </b><a NAME="E8:96_1_1_4_1"/>
<font color="Maroon">y</font> <a href="hidden.html#R2">in</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span>
 ;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E2:96_1_1_4"/>
 <a href="relat_1.html#K2">rng</a> <font color="Maroon">f</font> <a href="tarski.html#R1">c=</a> the <a href="struct_0.html#U1">carrier</a> of <span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span>
 <b>by </b><i><a class="ref" href="tarski.html#D3">TARSKI:def 3</a></i>;<br/>


</div><b>end;</b></div>
<b>then reconsider </b><font color="Maroon">f</font> = <font color="Maroon">f</font> as   <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">I</font>,<span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span> <b>by </b><i><a class="ref" href="quofield.html#D9" target="_self">Def9</a>, <a class="ref" href="funct_2.html#D1">FUNCT_2:def 1</a>, <a class="ref" href="relset_1.html#T11">RELSET_1:11</a></i>;<br/>
<a NAME="E7:96_1_1"/>
for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> holds  <font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span>
 
<div><a class="txt" onclick="hs2(this)" href="javascript:()" title="96_1_1_5"><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">I</font>;<br/>

<a NAME="E1:96_1_1_5"/>
<span class="p1"><a href="domain_1.html#K1">[</a><span class="default"><font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="quofield.html#K12" target="_self">QClass.</a> <span class="p3">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Maroon">x</font>,<span class="p4">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span></span>)</span></span><a href="domain_1.html#K1">]</a></span> <a href="hidden.html#R2">in</a> <font color="Maroon">f</font>
 
;<br/>
<b>hence </b><a NAME="E2:96_1_1_5"/>
<font color="Maroon">f</font> <a href="funct_2.html#K3">.</a> <font color="Maroon">x</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Maroon">x</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span>
 <b>by </b><i><a class="ref" href="quofield.html#D9" target="_self">Def9</a>, <a class="ref" href="funct_1.html#D4">FUNCT_1:def 4</a></i>;<br/>


</div><b>end;</b></div>
<b>hence </b><a NAME="E8:96_1_1"/>
 ex <font color="Olive">b<sub>1</sub></font> being  <a href="struct_0.html#NM6">Function</a> of <font color="Maroon">I</font>,<span class="p1">(<span class="default"><a href="quofield.html#K21" target="_self">the_Field_of_Quotients</a> <font color="Maroon">I</font></span>)</span> st <br/>for <font color="Olive">x</font> being  <a href="struct_0.html#NM1">Element</a> of <font color="Maroon">I</font> holds  <font color="Olive">b<sub>1</sub></font> <a href="funct_2.html#K3">.</a> <font color="Olive">x</font> <a href="hidden.html#R1">=</a>  <a href="quofield.html#K12" target="_self">QClass.</a> <span class="p1">(<span class="default"><a href="quofield.html#K23" target="_self">quotient</a> <font color="Olive">x</font>,<span class="p2">(<span class="default"><a href="group_1.html#K2">1.</a> <font color="Maroon">I</font></span>)</span></span>)</span>
 ;<br/>


</div>
